# Explicit synchronous partitioned scheme for coupled reduced order models
based on composite reduced bases

Amy de Castro<EMAIL_ADDRESS>Pavel Bochev<EMAIL_ADDRESS>Paul
Kuberry<EMAIL_ADDRESS>Irina Tezaur<EMAIL_ADDRESS>
###### Abstract

This paper formulates, analyzes and demonstrates numerically a method for the
explicit partitioned solution of coupled interface problems involving
combinations of projection-based reduced order models (ROM) and/or full order
models (FOMs). The method builds on the partitioned scheme developed in [1],
which starts from a well-posed formulation of the coupled interface problem
and uses its dual Schur complement to obtain an approximation of the interface
flux. Explicit time integration of this problem decouples its subdomain
equations and enables their independent solution on each subdomain. Extension
of this partitioned scheme to coupled ROM-ROM or ROM-FOM problems requires
formulations with non-singular Schur complements. To obtain these problems, we
project a well-posed coupled FOM-FOM problem onto a composite reduced basis
comprising separate sets of basis vectors for the interface and interior
variables, and use the interface reduced basis as a Lagrange multiplier. Our
analysis confirms that the resulting coupled ROM-ROM and ROM-FOM problems have
provably non-singular Schur complements, independent of the mesh size and the
reduced basis size. In the ROM-FOM case, analysis shows that one can also use
the interface FOM space as a Lagrange multiplier. We illustrate the
theoretical and computational properties of the partitioned scheme through
reproductive and predictive tests for a model advection-diffusion transmission
problem.

###### keywords:

partitioned scheme , projection-based reduced order model (ROM) , interface ,
transmission problem, inf-sup condition , proper orthogonal decomposition
(POD) , Galerkin method

††journal: Computer Methods in Applied Mechanics and
EngineeringSCSCfootnotetext: Department of Mathematics, Clemson University,
USACCRCCRfootnotetext: Center for Computing Research, Sandia National
Laboratories, Albuquerque, NM 87185, USASNLCASNLCAfootnotetext: Quantitative
Modeling & Software Engineering Department, Sandia National Laboratories, P.O.
Box 969, Livermore, CA 94551-0969, USAsand-blurbsand-blurbfootnotetext: This
paper describes objective technical results and analysis. Any subjective views
or opinions that might be expressed in the paper do not necessarily represent
the views of the U.S. Department of Energy or the United States Government.
Sandia National Laboratories is a multimission laboratory managed and operated
by National Technology and Engineering Solutions of Sandia, LLC., a wholly
owned subsidiary of Honeywell International, Inc., for the U.S. Department of
Energy’s National Nuclear Security Administration under contract DE-
NA-0003525.

##  1 Introduction

Partitioned methods are an attractive alternative to monolithic approaches for
both single and multi-physics applications. In the first case such schemes can
increase the concurrency of the simulation, improving computational
efficiency, by using an artificial interface to split the computational domain
into several subdomains. In the second case, where the interface is physical,
partitioned schemes enable both increased concurrency and reuse of existing
codes for the constituent physics components; see, e.g., [2] for an expository
survey. Because each individual component is solved independently, the codes
can run at their “sweet spots” utilizing, e.g., multi-rate time integrators
[3]. Performance of partitioned schemes in both simulation contexts can be
further enhanced by replacing the full-fidelity models in one or more
subdomains by computationally efficient projection-based reduced order models
(ROMs).

This work continues our efforts in [4] to extend the partitioned schemes in
[1] and [5] to interface problems in which a projection-based ROM on one of
the subdomains is coupled to either a full order model (FOM) or a ROM on the
other subdomain. In [4], we defined the subdomain ROMs by utilizing full
subdomain bases obtained by performing proper orthogonal decomposition (POD)
[6, 7] on a collection of snapshots containing both the interior and interface
degrees of freedom (DoFs). While this strategy is common in applications of
domain decomposition ideas to ROM (see, e.g., [8]), it does not guarantee that
the dual Schur complement system for the Lagrange multiplier is non-singular.
Unique solvability of this system is essential for the extension of the
partitioned schemes in [1] and [5] because the Lagrange multiplier defines a
Neumann boundary condition on the interface, which allows us to obtain well-
posed subdomain equations that can be solved independently.

The main contribution of this paper is the formulation and analysis of an
alternative approach which utilizes a _composite_ reduced basis, comprising
independently constructed ROM bases for the interfacial and interior DoFs,
instead of the conventional full subdomain reduced basis. To couple two ROMs
across an interface, we then use the _interface_ part of the composite basis
as a reduced order Lagrange multiplier space to enforce the interface
conditions. Our analysis reveals that this approach leads to a provably non-
singular Schur complement, independent of the underlying mesh size and/or
composite reduced basis dimension. For the coupling of a ROM to a FOM,
represented by a finite element model (FEM), this analysis indicates that one
can use either the interface part of the composite basis from the ROM side or
the interface finite element space from the FEM side of the interface as a
Lagrange multiplier. We provide numerical results that corroborate numerically
our theoretical findings. Results are shown on a two-dimensional (2D) time-
dependent advection-diffusion problem in the advection-dominated (high Péclet)
regime.

###  1.1 Related work

During the past two decades, the idea of coupling projection-based ROMs with
each other and with FOMs has been explored by a number of authors. The bulk of
the literature presents ROM-ROM or ROM-FOM coupling as a means for “gluing” or
“tiling” these ROMs and/or FOMs together. The focus is hence primarily on
using domain decomposition (DD) as a vehicle to _improve the efficiency of
model order reduction_ (MOR) for extreme scale problems and decomposable
problems. The coupling approaches in the literature fall into roughly two
categories: (1) monolithic coupling methods, and (2) iterative coupling
methods. We succinctly review the literature on both method categories in
Sections 1.1.1 and 1.1.2, respectively, and then in Section 1.2 we highlight
the key distinctions and contributions of this work.

####  1.1.1 Monolithic coupling methods

The majority of monolithic coupling methods in the MOR community employ
Lagrange multipliers to enforce compatibility constraints. Among the earliest
works exploring DD to perform coupling of POD-based ROMs in an effort to
improve the predictive accuracy is the work of Lucia et al. [9]. Another early
monolithic method for DD-based coupling of ROMs, this time constructed using
the Reduced Basis Element (RBE) method, is the work Maday et al. [8, 10].
These methods are different from ours in that they rely on Lagrange
multipliers represented by low-order polynomials (vs. POD modes) for imposing
compatibility in a mortar-type method that “glues” together non-overlapping
subdomains. In [11], Wicke et al. present an approach for stitching together
“composable” ROM “tiles”, precomputed given specific boundary conditions, with
the promise that the tiles can be assembled in arbitrary ways at runtime.
Continuity between tiles is enforced by duplicating the DoFs on the interfaces
and constraining their normal components to be equal. The Reduced basis method
with DD and Finite elements (RDF) [12] is a conceptually similar, non-
overlapping DD approach for gluing together networks of repetitive blocks. RDF
uses standard finite element bases on the interfaces between the ROM domains
to both enforce continuity and provide a sort of finite element enrichment. In
[13], Hoang et al. present an algebraically non-overlapping method for
coupling Least-Squares Petrov-Galerkin (LSPG) ROMs with each other. This
approach shares some commonality with the method developed herein, in that it
considers several different types of subdomain and interface bases. Unlike our
approach, [13] explores the use of both strong or weak compatibility
conditions imposed at the subdomain interfaces using Lagrange multipliers.

It is also possible to effect monolithic couplings without Lagrange
multipliers through a judicious construction of the underlying discrete
solution spaces. While these formulations are fundamentally different from our
Lagrange multiplier-based approach, we succinctly review several methods
falling into this category here for completeness. The works [12] and [14]
propose monolithic DD-based strategies for ROM-ROM and FOM-ROM coupling via
local reduced order bases, which are carefully constructed to ensure automatic
solution continuity across different subdomains. Another recent work that
accomplishes monolithic ROM-FOM coupling without Lagrange multipliers is [15].
Unlike our approach, which does not require that any specific discretization
method be used to discretize the governing PDE in space, the method in [15] is
based on a discontinuous Galerkin (DG) formulation, in which coupling is
achieved through the definition of numerical fluxes at discrete cell
boundaries.

It is interesting to remark that several DD-based ROM-ROM and ROM-FOM coupling
methods with on-the-fly basis and/or DD adaptation have been proposed in
recent years. While online model adaptation goes beyond the scope of the
present manuscript, it may be considered in a future publication. In [16] and
[17], Corigliano et al. develop a non-overlapping Lagrange multiplier-based
coupling method for nonlinear elasto-plastic and multi-physics problems, in
which on-the-fly ROM adaptation and ROM/FOM switching is performed through a
plastic check during the reduced analysis. Hybrid ROM-FOM coupling in the
context of solid mechanics applications is considered also in [18, 19], where
a local/global model reduction strategy for the simulation of quasi-brittle
fracture is developed. An adaptive sub-structuring (domain decomposition) non-
overlapping approach for ROM-ROM and ROM-FOM coupling in solid mechanics is
also presented in [20]. This method not only enables on-the-fly adaptation of
the ROM basis, but also on-the-fly substructuring/DD changes. Furthermore, in
the pre-print [21], Huang et al. develop a component-based modeling framework
that can flexibly integrate ROMs and FOMs for different components or domain
decompositions, towards modeling accuracy and efficiency for complex, large-
scale combustion problems. It is demonstrated that accuracy can be enhanced by
incorporating basis adaptation ideas from [22, 23].

Finally, it is worth mentioning that another recent direction for hybrid ROM-
FOM and ROM-ROM coupling involves the integration of ideas from machine
learning into the coupling formulation. For example, in [24], Ahmed et al.
present a hybrid ROM-FOM approach in which a long short-term memory network is
introduced at the interface and subsequently used to perform the multi-model
coupling.

####  1.1.2 Iterative coupling methods

While iterative coupling methods are fundamentally different from the
monolithic couplings developed in the present work, we overview several recent
efforts falling into the iterative coupling category here for completeness.
The majority of iterative methods for ROM-ROM and ROM-FOM coupling are based
on the Schwarz alternating method [25]. Iterative coupling methods have the
advantage that they are often less intrusive to implement in existing high-
performance computing (HPC) codes [26, 27]; however, the methods’ iterative
nature can add to the total CPU time required to complete a simulation.

Among the earliest Schwarz-based DD approaches for coupling FOMs with ROMs is
the work of Buffoni et al. [28], which focuses on Galerkin-free POD ROMs
developed for the Laplace equation and the compressible Euler equations.
Galerkin-free FOM-ROM and ROM-ROM couplings are also considered by Cinquegrana
et al. [29] and Bergmann et al. [30]. The former approach [29] considers
overlapping DD in the context of a Schwarz-like iteration scheme, but, unlike
our approach, requires matching meshes at the subdomain interfaces. The latter
approach [30], termed zonal Galerkin-free POD, defines an optimization problem
which minimizes the difference between the POD reconstruction and its
corresponding FOM solution in the overlapping region between a ROM and a FOM
domain. A true POD-Greedy/Galerkin non-overlapping Schwarz method for the
coupling of projection-based ROMs developed for the specific case of symmetric
elliptic PDEs is presented by Maier et al. in [31]. A more general
POD/Galerkin ROM-ROM and ROM-FOM coupling method based on the overlapping or
non-overlapping Schwarz method is developed in [32]. This work is an extension
of an alternating Schwarz-based concurrent multi-scale FOM-FOM coupling method
developed earlier in [26, 27]. A recent work by Iollo et al. [33] on
component-based model reduction via overlapping alternating Schwarz shows
that, for linear elliptic PDEs, the latter can be interpreted as an
optimization-based coupling [34]. By solving the optimization problem
directly, one obtains a “One-Shot Schwarz” procedure.

Alternative ROM-FOM approaches include the domain decomposition non-intrusive
reduced-order model (DDNIR) [35]. Here, a radial basis function interpolation
method is used to construct a set of hypersurfaces for iterative solution
transfer between neighboring subdomains.

While our focus is restricted to projection-based ROMs, it is worth noting
that the Schwarz alternating method has recently been extended to DD coupling
of Physics Informed Neural Networks (PINNs) in [36, 37]. The methods proposed
in these works, termed D3M [37] and DeepDDM [36], inherit the benefits of DD-
based ROM-ROM couplings, but are developed primarily for the purpose of
improving the efficiency of the neural network training process and reducing
the risk of over-fitting, both of which are due to the global nature of the
neural network “basis functions”. The Schwarz alternating method has also been
used for online coupling of independently pre-trained subdomain-localized
neural network-based models, e.g. in [38], which develops a transferable
framework for solving boundary value problems (BVPs) via deep neural networks
that can be trained once and used forever for various unseen domains and
boundary conditions (BCs).

###  1.2 Differentiating contributions and organization

The partitioned schemes for coupled ROM-ROM and ROM-FOM problems in this paper
have some commonality with both the monolithic Lagrange multiplier-based
coupling approaches described succinctly in Section 1.1.1, e.g., [8, 10, 12,
39, 13], and the iterative coupling schemes summarized in Section 1.1.2 in the
sense that they all focus on couplings between ROMs and/or ROMs and FOMs.
However, the work presented here differs from both of these types of methods
in several important ways.

Compared to the methods in Section 1.1.1, our main focus is on _improving the
simulation efficiency_ for both single and multi-physics problems through
_explicit partitioned solution_ of their coupled ROM-ROM and ROM-FOM
formulations, rather than on _improving the efficiency of the model order
reduction_ process through domain-decomposition ideas. Second, although both
DD-ROM methods and our partitioned scheme utilize Lagrange multipliers, the
variational setting for the coupled ROM-ROM and ROM-FOM formulations in this
paper differs from the one in a typical DD-based ROM; this is because ours is
designed to provide a provably non-singular Schur complement when the coupled
ROM-ROM or ROM-FOM problems are discretized by an explicit time integrator. As
explained in Remark 7, this makes our setting more “forgiving” to variational
crimes and results in Schur complements whose conditioning is independent of
the mesh size underpinning the FOM or the size of the composite reduced basis
defining the ROM.

Insofar as the iterative coupling methods are concerned, both our partitioned
schemes and the methods utilizing the Schwarz alternating algorithm perform
independent solves of decoupled subdomain problems. Additionally, in both
cases, the decoupling is effected by specifying boundary conditions on the
interface that “close” the subdomain equations and make their independent
solution possible. However, in the case of the Schwarz alternating method, one
usually starts with an initial guess for the boundary condition and iterates
until the subdomain solutions have converged sufficiently. The rate of
convergence generally depends on the size of the overlap between the
subdomains, which makes this type of methods more difficult to extend to
multiphysics problems where different subdomains may have different sets of
governing equations. In contrast, our partitioned schemes define the interface
boundary conditions by solving a Schur complement equation that provides a
highly accurate estimate of the interface flux. Conceptually, this approach is
similar to the techniques in [40] where one solves an additional problem to
obtain more accurate approximation of the boundary flux than afforded by
simply inserting the finite element solution into the flux function.

The remainder of this paper is organized as follows. Section 2 introduces the
bulk of the notation used in the paper. In Section 3, we describe our model
transmission problem (a transient scalar advection-diffusion problem) and
define the coupled FOM-FOM formulation, which provides the basis for the
development of the coupled ROM-ROM and ROM-FOM problems. For convenience, in
Section 3.2, we also briefly summarize the partitioned Implicit Value Recovery
(IVR) scheme [1]. Section 4 overviews projection-based model reduction using
the POD/Galerkin method and introduces the reduced order basis spaces that
will be used in the paper. Section 5 is the core of this paper, where we use
the composite reduced basis idea to formulate the IVR scheme for the
partitioned solution of two ROMs coupled across an interface. Next, in Section
6, we describe how the IVR scheme can be extended to the partitioned solution
of coupled ROM-FOM problems. In Section 7, we use variational techniques to
prove that the coupled ROM-ROM and ROM-FOM formulations have non-singular
Schur complements, which is the key prerequisite for the extension of the IVR
scheme to these couplings. Numerical results demonstrating the proposed
scheme’s accuracy and efficiency for reproductive as well as predictive ROMs
are presented in Section 8. Finally, conclusions are offered in Section 9.

##  2 Notation

Figure 1: Example partitioning of 2D domain $\Omega$ into two non-overlapping
subdomains, $\Omega_{1}$ and $\Omega_{2}$, with interface boundary $\gamma$.

For the convenience of the reader and ease of reference, this section
summarizes the bulk of the notation used throughout the paper. We consider a
bounded region $\Omega\in\mathbb{R}^{\nu}$, $\nu=2,3$ divided into two111This
configuration of the model transmission problem possesses all the
characteristics relevant for the development of the partitioned schemes.
Extension of these schemes to transmission problems with more than two domains
is conceptually similar to the case of two subdomains. non-overlapping
subdomains $\Omega_{1}$ and $\Omega_{2}$ by an interface $\gamma$, as shown in
Figure 1. Without a loss of generality, we assume that the unit normal
$\bm{n}_{\gamma}$ points towards $\Omega_{2}$, and set
$\Gamma_{i}:=\partial\Omega_{i}\backslash\gamma$, $i=1,2$.

We use the standard notation $L^{2}(\Omega_{i})$ for the space of all square
integrable functions in $\Omega_{i}$ with norm and inner product denoted by
$\|\cdot\|_{0,\Omega_{i}}$ and $(\cdot,\cdot)_{0,\Omega_{i}}$, respectively.
Likewise, ${H}^{1}(\Omega_{i})$ will denote the Sobolev space of all square
integrable scalar functions on $\Omega_{i}$ whose first derivatives are also
square integrable and ${H}^{1}_{D}(\Omega_{i})$ will be the subspace of
${H}^{1}(\Omega_{i})$ whose elements vanish on $\Gamma_{i}$. Restrictions of
${H}^{1}(\Omega_{i})$ functions to $\gamma$ form the fractional Sobolev space
$H^{1/2}(\gamma)$, with dual $H^{-1/2}(\gamma)$ and duality pairing
$\langle\cdot,\cdot\rangle_{\gamma}$.

In this paper we consider quasi-uniform partitions $\Omega^{h}_{i}$ of
$\Omega_{i}$ with mesh parameter $h_{i}$, vertices $\bm{x}_{i,r}$, and
elements $K_{i,s}$. We assume that each subdomain is meshed independently and
denote the finite element partition of the interface induced by
$\Omega^{h}_{i}$ as $\gamma^{h}_{i}$. To avoid technicalities that are not
germane to the subject of this paper, we shall assume that $\gamma^{h}_{1}$
and $\gamma^{h}_{2}$ are spatially coincident; however, their vertices are not
required to match. Similarly, $\Gamma^{h}_{i}$ will be the finite element
partition of the Dirichlet boundary induced by $\Omega^{h}_{i}$.

######  Remark 1.

If a node $\bm{x}_{i,r}$ lies on both $\Gamma_{i}$ and $\gamma$, it is treated
as belonging to the finite element partition $\Gamma^{h}_{i}$ of the Dirichlet
boundary rather than that of the interface; see Figure 2. The reason for this
is that the DoFs at these nodes are assigned the nodal values of the boundary
data and they contribute to the right-hand sides of the discrete equations,
i.e., these DoFs are not unknown solution coefficients.

Figure 2: Node partitions of the finite element mesh. Solid disks depict
Dirichlet nodes forming the space $S^{h}_{i,\Gamma}$; circles depict interior
and interface nodes forming the space $S^{h}_{i,D}$; diamonds depict interface
nodes forming the spaces $S^{h}_{i,\gamma}$ and $G^{h}_{i}$. Arrows indicate
duplicate nodes that correspond to the same DoFs.

In what follows, $S_{i}^{h}$ will denote the lowest-order nodal $C^{0}$
conforming finite element subspace of ${H}^{1}(\Omega_{i})$, defined with
respect to $\Omega^{h}_{i}$; see, e.g., [41]. We equip $S_{i}^{h}$ with a
standard Lagrangian basis $\\{N_{i,r}\\}$, i.e.,
$N_{i,r}(\bm{x}_{i,s})=\delta_{rs}$, where $\delta_{rs}$ is the Kronecker
$\delta$-symbol. We denote the subspace of $S_{i}^{h}$ comprising all finite
element functions that vanish on $\Gamma_{i}$ by $S_{i,D}^{h}$. This subspace
is a conforming approximation of the Sobolev space ${H}^{1}_{D}(\Omega_{i})$.
Let $n_{i,\Gamma}$, $n_{i,\gamma}$ and $n_{i,0}$ denote the numbers of mesh
nodes on the Dirichlet boundary $\Gamma_{i}$, the interface $\gamma$, and the
interior of the subdomain $\Omega_{i}$, respectively; see Figure 2. Thus,
$n_{i,D}=n_{i,0}+n_{i,\gamma}$ and
$n_{i}=n_{i,D}+n_{i,\Gamma}=n_{i,0}+n_{i,\gamma}+n_{i,\Gamma}$.

The coefficients of a finite element function $u^{h}_{i}\in S_{i}^{h}$ form a
vector $\bm{u}_{i}\in\mathbb{R}^{n_{i}}$. Without loss of generality, we can
assume that the nodes of $\Omega^{h}_{i}$ are numbered such that this
coefficient vector has the form
$\bm{u}_{i}=(\bm{u}_{i,\gamma},\bm{u}_{i,0},\bm{u}_{i,\Gamma})$, where
$\bm{u}_{i,\gamma}\in\mathbb{R}^{n_{i,\gamma}}$,
$\bm{u}_{i,0}\in\mathbb{R}^{n_{i,0}}$, and
$\bm{u}_{i,\Gamma}\in\mathbb{R}^{n_{i,\Gamma}}$ are vectors of interface,
interior, and Dirichlet coefficients, respectively. With this convention, it
is easy to see that the coefficient vector of a finite element function
$u^{h}_{i,D}\in S^{h}_{i,D}$ has the form
$\bm{u}_{i,D}=(\bm{u}_{i,\gamma},\bm{u}_{i,0},\bm{0})$.

Besides $S^{h}_{i,D}$, we shall need three additional subspaces of
$S^{h}_{i}$. The first one contains all finite element functions whose
coefficient vectors have the form $(\bm{u}_{i,\gamma},\bm{0},\bm{0})$, i.e.,
they vanish at all but the interface nodes. We denote this space by
$S^{h}_{i,\gamma}$ and call it the _interface_ part of $S^{h}_{i}$. The finite
element functions in the second subspace have coefficient vectors
$(\bm{0},\bm{u}_{i,0},\bm{0})$. These functions are identically zero on both
the interface and the Dirichlet boundary. We term this subspace the _interior_
part of $S^{h}_{i}$ and denote it by $S^{h}_{i,0}$. The last subspace of
$S^{h}_{i}$ contains all finite element functions with coefficients
$(\bm{0},\bm{0},\bm{u}_{i,\Gamma})$. These functions vanish at all nodes
except those on the Dirichlet boundary $\Gamma_{i}$. We denote this space by
$S^{h}_{i,\Gamma}$ and call it the _boundary_ part of $S^{h}_{i}$. Note that

$S^{h}_{i}=S_{i,\gamma}^{h}\cup S_{i,0}^{h}\cup S^{h}_{i,\Gamma}\
\quad\mbox{and}\quad S^{h}_{i,D}=S_{i,\gamma}^{h}\cup S_{i,0}^{h}\,.$

Formally, the subspaces $S^{h}_{i,D}$, $S^{h}_{i,\gamma}$, $S^{h}_{i,0}$, and
$S^{h}_{i,\Gamma}$ have the same dimension $n_{i}$ as their parent space
$S^{h}_{i}$. However, when discussing the assembled algebraic forms of the
weak formulations, it will be more convenient to remove the zero blocks from
the coefficient vectors and associate $u^{h}_{i,D}\in S^{h}_{i,D}$,
$u^{h}_{i,\gamma}\in S^{h}_{i,\gamma}$, $u^{h}_{i,0}\in S^{h}_{i,0}$, and
$u^{h}_{i,\Gamma}\in S^{h}_{i,\Gamma}$ with coefficient vectors
$\bm{u}_{i,D}\in\mathbb{R}^{n_{i,D}}$,
$\bm{u}_{i,\gamma}\in\mathbb{R}^{n_{i,\gamma}}$,
$\bm{u}_{i,0}\in\mathbb{R}^{n_{i,0}}$, and
$\bm{u}_{i,\Gamma}\in\mathbb{R}^{n_{i,\Gamma}}$, respectively. In this
context, we will refer to $n_{i,D}$, $n_{i,\gamma}$, $n_{i,0}$, and
$n_{i,\Gamma}$ as the _effective_ dimensions of their respective finite
element subspaces. Finally, we define the _induced_ interface finite element
space $G_{i}^{h}$ as the trace of the interface part of $S^{h}_{i}$, i.e.,
$G_{i}^{h}=S_{i,\gamma}^{h}\big{|}_{\gamma}$. Since
$dim\,G_{i}^{h}=dim\,S_{i,\gamma}^{h}=n_{i,\gamma}$, the coefficient spaces of
$G_{i}^{h}$ and $S_{i,\gamma}^{h}$ are isomorphic, i.e., a vector
$\bm{c}\in\mathbb{R}^{n_{i,\gamma}}$ can be mapped to a function
$u^{h,\bm{c}}_{i,\gamma}\in S_{i,\gamma}^{h}$, or a function
$\lambda^{h,\bm{c}}_{i}\in G^{h}_{i}$.

##  3 Model problem and its coupled FOM-FOM formulation

This section defines the model transmission problem, the associated weak
coupled formulation and its semi-discretization in space222We emphasize that,
while the high-fidelity models herein are assumed to be constructed using the
finite element method, our partitioned solution approach is easily extensible
to FOMs constructed using alternate discretization approaches such as finite
volume and finite difference methods. by finite elements. The coupled FOM-FOM
problem is key to the development of partitioned schemes for coupled ROM-ROM
and ROM-FOM problems in this paper. For completeness, Section 3.2 briefly
reviews the IVR scheme for the FOM-FOM problem.

We consider the scalar advection-diffusion transmission equation

$\displaystyle\begin{split}\dot{u}_{i}-\nabla\cdot F_{i}(u_{i})&=f_{i}\hskip
14.22636pt\text{ on }\Omega_{i}\times[0,T]\\\ u_{i}&=g_{i}\hskip
14.22636pt\text{ on }\Gamma_{i}\times[0,T]\\\ u_{i}&=u_{i,0}\hskip
5.69054pt\text{ on }\Omega_{i},\hskip 5.69054ptt=0\end{split}\;\quad i=1,2,$
(1)

where the over-dot notation denotes differentiation in time, the unknown
$u_{i}:=u_{i}(\bm{x},t)$ is a scalar field, $F_{i}(u_{i})=\kappa_{i}\nabla
u_{i}-\mathbf{a}u_{i}$ is the total flux function, $f_{i}:=f_{i}(\bm{x},t)$ is
a source term, $g_{i}:=g_{i}(\bm{x},t)$ is prescribed boundary data,
$u_{i,0}:=u_{i,0}(\bm{x})$, is a prescribed initial condition,
$\kappa_{i}:=\kappa_{i}(\bm{x},t)>0$ is the diffusion coefficient in
$\Omega_{i}$, and $\mathbf{a}:=\mathbf{a}(\bm{x},t)$ is the advection field.
Along the interface $\gamma$, we enforce continuity of the “velocities”
$\dot{u}_{i}$ and continuity of the total flux, giving rise to the following
interface conditions:

$\displaystyle\dot{u}_{1}(\bm{x},t)-\dot{u}_{2}(\bm{x},t)=0\hskip
5.69054pt\text{ and
}F_{1}(\bm{x},t)\cdot\mathbf{n}_{\gamma}=F_{2}(\bm{x},t)\cdot\mathbf{n}_{\gamma}\hskip
5.69054pt\text{ on }\gamma\times[0,T].$ (2)

We choose this problem because it allows us to conveniently demonstrate the
partitioned methods developed in this paper in both simulation contexts by
setting $\kappa_{1}=\kappa_{2}$ and $\kappa_{1}\neq\kappa_{2}$, respectively.

######  Remark 2.

The use of “velocity” continuity in lieu of the more conventional continuity
of the states coupling condition is required to obtain a coupled FOM-FOM
formulation that, under some conditions on the Lagrange multiplier space, is a
Hessenberg Index-1 Differential Algebraic Equation (DAE) [42]. In such DAEs
the algebraic variable (the Lagrange multiplier) is an implicit function of
the differential variables (the subdomain states). This fact is at the core of
the IVR formulation as it allows one to solve for the Lagrange multiplier in
terms of the subdomain states. It also motivates the term “implicit” in the
name of the scheme.

We define the coupled weak formulation of (1) by using a Lagrange multiplier
to enforce the first constraint in (2). To that end, we write the solution as
$u_{i}=u_{i,D}+g_{i}$ where $u_{i,D}\in{H}^{1}_{D}(\Omega_{i})$ is the
interior (unknown) component of $u_{i}$ and, with some abuse of notation,
$g_{i}\in{H}^{1}(\Omega_{i})$ is a lifting of the boundary data. The weak form
of (1)–(2) is then given by the variational equation: _seek_
$\\{u_{1,D},u_{2,D},\lambda\\}:(0,T]\mapsto{H}^{1}_{D}(\Omega_{1})\times{H}^{1}_{D}(\Omega_{2})\times{H}^{-1/2}(\gamma)$
_such that_ $u_{i}=u_{i,0}$ for $t=0$, $i=1,2$, and for $t>0$

$\begin{array}[]{rcll}\left(\dot{u}_{1,D},v_{1}\right)_{0,\Omega_{1}}+\left\langle\lambda,v_{1}\right\rangle_{\gamma}&=&\left(f_{1},v_{1}\right)_{0,\Omega_{1}}-\left(F_{1}(u_{1,D}),\nabla
v_{1}\right)_{0,\Omega_{1}}-Q_{1}(\dot{g}_{1},g_{1};v_{1})&\quad\forall
v_{1}\in H^{1}_{\Gamma}(\Omega_{1})\\\\[4.30554pt]
\left(\dot{u}_{2,D},v_{2}\right)_{0,\Omega_{2}}-\left\langle\lambda,v_{2}\right\rangle_{\gamma}&=&\left(f_{2},v_{2}\right)_{0,\Omega_{2}}-\left(F_{2}(u_{2,D}),\nabla
v_{2}\right)_{0,\Omega_{2}}-Q_{2}(\dot{g}_{2},g_{2};v_{2})&\quad\forall
v_{2}\in H^{1}_{\Gamma}(\Omega_{2})\\\\[4.30554pt]
\left\langle\dot{u}_{1,D}-\dot{u}_{2,D},\mu\right\rangle_{\gamma}&=&\left\langle\dot{g}_{1}-\dot{g}_{2},\mu\right\rangle_{\gamma}&\quad\forall\mu\in
H^{-1/2}(\gamma).\end{array}$ (3)

In (3), the $Q_{i}$, for $i=1,2$, denote bilinear forms producing the
contributions from the boundary data to the right-hand sides of the subdomain
equations. The variational equation (3) is of the mixed type. Using the theory
in [43], one can show that (3) is well-posed.

######  Remark 3.

Since the Dirichlet data are supposed to satisfy the first coupling condition
in (2), the boundary data contribution
$\left\langle\dot{g}_{1}-\dot{g}_{2},\mu\right\rangle_{\gamma}$ to the right-
hand side of the constraint equation in (3) is identically zero. However, in
general, a discretized version of this term will not be identically zero and
boundary contributions need to be properly accounted for in the assembled
discrete problem.

###  3.1 The coupled FOM-FOM problem

To obtain the coupled FOM-FOM problem, we discretize the weak formulation (3)
in space by approximating the subdomain states $\\{u_{1},u_{2}\\}$ and the
Lagrange multiplier $\lambda$ with the finite element spaces
$V^{h}=S^{h}_{1}\times S^{h}_{2}$ and $W^{h}=G^{h}_{k}$, $k=1$ or $k=2$,
respectively. This choice of $W^{h}$ is common for mortar element methods [44,
45] and it also ensures the well-posedness of the IVR scheme for the FOM-FOM
problem.

To handle the Dirichlet boundary conditions, we proceed similarly to (3) and
write the finite element solution as $u^{h}_{i}=u^{h}_{i,D}+g^{h}_{i}$, where
$u^{h}_{i,D}\in S^{h}_{i,D}$ is the unknown part of $u^{h}_{i}$ and
$g^{h}_{i}\in S^{h}_{i,\Gamma}$ is the finite element interpolant of the
boundary data. Thus, the coefficient vector of $u^{h}_{i}$ is given by
$\bm{u}_{i}=(\bm{u}_{i,D},\bm{g}_{i})$ where
$\bm{u}_{i,D}\in\mathbb{R}^{n_{i,D}}$ is the coefficient vector containing the
unknown nodal values of the solution, and
$\bm{g}_{i}\in\mathbb{R}^{n_{i,\Gamma}}$ is a coefficient vector containing
the known nodal values of the boundary data $g_{i}$. To obtain the coupled
FOM-FOM problem, we approximate the duality pairing
$\langle\cdot,\cdot\rangle_{\gamma}$ by the $L^{2}$ inner product333Remark 7
provides some additional information about this choice for the interface inner
product. $(\cdot,\cdot)_{0,\gamma}$ and restrict (3) to $V^{h}$ and $W^{h}$.
The resulting problem can be written in the following compact matrix form:

$\displaystyle\begin{bmatrix}M_{1,D}&0&G_{1,D}^{T}\\\\[4.30554pt]
0&M_{2,D}&-G_{2,D}^{T}\\\\[4.30554pt]
G_{1,D}&-G_{2,D}&0\end{bmatrix}\begin{bmatrix}\bm{\dot{u}}_{1,D}\\\\[4.30554pt]
\bm{\dot{u}}_{2,D}\\\\[4.30554pt]
\bm{\lambda}\end{bmatrix}=\begin{bmatrix}\bm{f}_{1,D}-F_{1,D}\bm{u}_{1,D}-Q_{1,\Gamma}(\dot{\bm{g}}_{1},\bm{g}_{1})\\\\[4.30554pt]
\bm{f}_{2,D}-F_{2,D}\bm{u}_{2,D}-Q_{2,\Gamma}(\dot{\bm{g}}_{2},\bm{g}_{2})\\\\[4.30554pt]
-Q_{\gamma,\Gamma}(\dot{\bm{g}}_{1},\bm{\dot{g}}_{2})\end{bmatrix},$ (4)

where, for $i=1,2$, $M_{i,D}$ and $F_{i,D}$ are $n_{i,D}\times n_{i,D}$ mass
and flux matrices, respectively, $G_{i,D}$ are $n_{k,\gamma}\times n_{i,D}$
matrices defining the algebraic form of the “velocity” constraint in (2),
$\bm{\lambda}\in\mathbb{R}^{n_{k,\gamma}}$ is the coefficient vector of the
discrete Lagrange multiplier, and $\bm{f}_{i,D}\in\mathbb{R}^{n_{i,D}}$ is the
coefficient vector of the source term. The terms

$Q_{i,\Gamma}(\dot{\bm{g}}_{i},\bm{g}_{i})=M_{i,\Gamma}\dot{\bm{g}}_{i}+F_{i,\Gamma}\bm{g}_{i},\quad
i=1,2\quad\mbox{and}\quad
Q_{\gamma,\Gamma}(\dot{\bm{g}}_{1},\bm{\dot{g}}_{2})=G_{1,\Gamma}\dot{\bm{g}}_{1}-G_{2,\Gamma}\bm{\dot{g}}_{2},$

where $M_{i,\Gamma}$ and $F_{i,\Gamma}$ are $n_{i,D}\times n_{i,\Gamma}$
“partial” mass and flux matrices, and $G_{i,\Gamma}$ are $n_{k,\gamma}\times
n_{i,\Gamma}$ “partial” constraint matrices, provide the contributions from
the boundary data interpolants to the right-hand sides of the subdomain
equations. Note that the matrix blocks in (4) are dimensioned using the
effective dimensions of the finite element spaces.

######  Remark 4.

The “partial” constraint matrices $G_{i,\Gamma}$ are, in general, very sparse.
For example, in two-dimensions, each $G_{i,\Gamma}$ will have at most two non-
zero elements corresponding to the two Dirichlet nodes at the endpoints of
$\gamma$; see Figure 2. Although $\dot{\bm{g}}_{1}=\dot{\bm{g}}_{2}$ at these
nodes, the integrals of the finite element interpolants $\dot{g}^{h}_{1}$ and
$\dot{g}^{h}_{2}$ against the Lagrange multiplier basis functions will not be
identical unless the nodes adjacent to the endpoints of $\gamma$ match on both
sides of the interface. This is to be contrasted with the continuous problem
where the Dirichlet data does not contribute to the constraint equation; see
Remark 3.

###  3.2 The Implicit Value Recovery (IVR) scheme for the coupled FOM-FOM
problem

In this section, we briefly review the IVR scheme [1] for the coupled FOM-FOM
problem (4). This scheme solves the linear system

$\begin{array}[]{rcll}S\bm{\lambda}&=&G_{1,D}M_{1,D}^{-1}\left[\bm{f}_{1,D}-F_{1,D}\bm{u}_{1,D}-Q_{1,\Gamma}(\dot{\bm{g}}_{1},\bm{g}_{1})\right]\\\
&&\;-G_{2,D}M_{2,D}^{-1}\left[\bm{f}_{2,D}-F_{2,D}\bm{u}_{2,D}-Q_{2,\Gamma}(\dot{\bm{g}}_{2},\bm{g}_{2})\right]+Q_{\gamma,\Gamma}(\bm{\dot{g}}_{1},\bm{\dot{g}}_{2}),\end{array}$
(5)

where

$S=G_{1,D}M^{-1}_{1,D}G^{T}_{1,D}+G_{2,D}M^{-1}_{2,D}G^{T}_{2,D}$ (6)

is the dual Schur complement of the matrix on the left hand side of (4). This
is used to compute a highly accurate approximation of the interface flux444In
contrast, “loosely coupled” partitioned schemes use the “raw” solution state
from each side of the interface to specify boundary conditions that close each
subdomain equation and make possible its independent solution. Mathematically,
such schemes can be viewed as performing a single step of a non-overlapping
alternating Schwarz iterative coupling procedure; see Section 1.1.2. This is
also the root cause for some of the stability and accuracy issues experienced
by these methods. $\bm{\lambda}$, which then serves as a Neumann boundary
condition for the subdomain equations. As a result, the well-posedness of the
IVR scheme hinges on the invertibility of the Schur complement matrix $S$. A
sufficient condition for (6) to be symmetric and positive definite is that the
transpose constraint matrix has a full column rank. One can show that if the
Lagrange multiplier space $W^{h}$ is defined as in Section 3.1, i.e., as the
trace $G^{h}_{k}$ of the interface finite element space $S^{h}_{k,\gamma}$ on
either of $\Omega_{1}$ or $\Omega_{2}$, the matrix
$G^{T}=(G_{1,D},-G_{2,D})^{T}$ does indeed have this property.

Assuming a non-singular Schur complement (6), the IVR scheme for (1) comprises
the following two steps. First, one solves (5) for the Lagrange multiplier
$\bm{\lambda}$ and eliminates it from (4). This reduces the coupled FOM-FOM
problem to a coupled system of two ordinary differential equations (ODEs):

$\displaystyle\begin{bmatrix}M_{1,D}&0\\\ 0&M_{2,D}\\\
\end{bmatrix}\begin{bmatrix}\bm{\dot{u}}_{1,D}\\\ \bm{\dot{u}}_{2,D}\\\
\end{bmatrix}=\begin{bmatrix}\bm{f}_{1,D}-F_{1,D}\bm{u}_{1,D}-G^{T}_{1,D}\bm{\lambda}(\bm{u}_{1,D},\bm{u}_{2,D};\dot{\bm{g}}_{1},\dot{\bm{g}}_{2})-Q_{1,\Gamma}(\dot{\bm{g}}_{1},\bm{g}_{1})\\\
\bm{f}_{2,D}-F_{2,D}\bm{u}_{2,D}+G^{T}_{2,D}\bm{\lambda}(\bm{u}_{1,D},\bm{u}_{2,D};\dot{\bm{g}}_{1},\dot{\bm{g}}_{2})-Q_{2,\Gamma}(\dot{\bm{g}}_{2},\bm{g}_{2})\\\
\end{bmatrix}\,.$ (7)

The ODE sub-systems in (7) define the associated _subdomain_ FOMs. It is easy
to see that an explicit time discretization of (7) decouples this problem and
allows one to advance the solution to the next time step by solving the
subdomain FOMs _independently_ ; see [1]. Thus, the second step of IVR
consists of applying explicit time integrators to each subdomain FOM. The
subdomain time integrators are not required to be the same and they can also
use different time steps over shared synchronization time intervals.

Because decoupling of (7) is effected solely by explicit time integration, it
is not accompanied by any splitting errors as is the case with traditional
loosely coupled partitioned schemes. In particular, the IVR scheme fully
retains the stability and the accuracy properties of the underlying coupled
problem. In fact, one can show that, for some settings, the IVR solution is
identical to the solution of the coupled problem.

##  4 Projection-based model order reduction (MOR)

In this section, we briefly review the basic concepts of the MOR approach used
in this paper to develop the IVR scheme for the partitioned solution of
coupled problems involving subdomain-local ROMs coupled to other subdomain-
local ROMs or to FOMs. We then specialize some aspects of the generic MOR
process to the model transmission problem that is the focus of this paper.

###  4.1 A generic POD-based MOR workflow

The approach for constructing a projection-based ROM consists of two critical
steps: (1) calculation of a reduced basis (RB), and (2) projection of the
governing equations onto the reduced basis. These two steps are described
succinctly in the following paragraphs.

##### Reduced basis construction via the POD

One of the most popular approaches for calculating a reduced basis is the POD
[7, 6]. To discuss POD, consider a generic FOM given by

$M\dot{\bm{u}}=\bm{f}(\bm{u}),$ (8)

where $M\in\mathbb{R}^{n\times n}$ and $\bm{u},\bm{f}\in\mathbb{R}^{n}$. The
FOM (8) can be thought of as resulting from a spatial discretization of some
set of governing PDEs.

To obtain the POD basis, one simulates (8) and collects its solutions
$\bm{u}^{m}$, $m=1,\ldots,r$ into an $n\times r$ snapshot matrix $X.$
Typically, the snapshots $\bm{u}^{m}$ are taken to be the primary solution
field at different times and/or different parameter values. POD works by first
computing the singular value decomposition (SVD) $X=U\Sigma V^{T}$ of the
snapshot matrix. Then, one chooses a positive integer $0<d\leq n$ that defines
the accuracy of the reduced basis. The value of $d$ is typically selected
using a “snapshot energy” criterion, where the “snapshot energy” is defined as

$\displaystyle\mathcal{E}:=\frac{\sum_{i=1}^{d}\sigma_{i}^{2}}{\sum_{i=1}^{n}\sigma_{i}^{2}},$
(9)

with $\sigma_{i}$ denoting the $i^{th}$ singular value of $X$. Specifically,
let $\delta$ be a desired threshold for the retention of the snapshot energy.
The integer $d$ is then defined as the smallest integer such that

$\sum_{i=1}^{d}\sigma_{i}^{2}\geq\Big{(}1-\delta\Big{)}\sum_{i=1}^{n}\sigma_{i}^{2}.$
(10)

Typically, one seeks a reduced basis that captures 95% or 99% of the snapshot
energy, i.e., $\mathcal{E}\approx 0.95$ or $\mathcal{E}\approx 0.99$. This
corresponds to thresholds $\delta=0.05$ or $\delta=0.01$.

Once $d$ is determined according to (10) the $n\times d$ POD reduced basis
matrix, denoted herein by ${{\color[rgb]{0,0,0}{\Phi}}}$, is defined by taking
the first $d$ left singular vectors of $X$, i.e., the first $d$ columns of
$U$. Construction of the POD basis can be interpreted as an approximation of
the snapshot set $X$ by its truncated SVD:
$X\approx\widetilde{X}={{\color[rgb]{0,0,0}{\Phi}}}\widetilde{\Sigma}\widetilde{V}^{T}$.
In order to achieve a meaningful order reduction of the FOM, $d$ must be much
smaller than the dimension $n$ of the FOM. We remark that this requires a
sharp decay of the singular values, which holds for our model problem but is
not true in general for problems with a slow decay of the so-called Kolmogorov
$n$-width [46]; see, e.g., [47, 48, 49].

Once the reduced basis ${{\color[rgb]{0,0,0}{\Phi}}}$ is calculated using the
above workflow, the FOM solution $\bm{u}$ is approximated as a linear
combination of these reduced basis modes and unknown time-dependent modal
amplitudes $\widetilde{\bm{u}}\in\mathbb{R}^{d}$:

$\bm{u}(t)\approx\bar{\bm{u}}+{{\color[rgb]{0,0,0}{\Phi}}}\widetilde{\bm{u}}(t).$
(11)

In (11), $\bar{\bm{u}}\in\mathbb{R}^{n}$ is a reference state, commonly
selected as the initial condition, base flow or snapshot mean. An important
detail in our formulation is that $\bar{\bm{u}}$ can be a function of time,
i.e., $\bar{\bm{u}}:=\bar{\bm{u}}(t)$. As we show below, $\bar{\bm{u}}$ can
also be used to enforce time-varying Dirichlet boundary conditions strongly
within the POD-based ROM.

##### Galerkin projection

Given a reduced basis ${{\color[rgb]{0,0,0}{\Phi}}}$, the next step is to
project the FOM onto this basis. Here, we restrict our discussion to an
approach called “discrete Galerkin projection”, where the governing equations
in their semi-discretized form (8) (i.e., discretized only in space) are
projected onto the POD basis in the discrete $l^{2}$ inner product. Projecting
(8) onto a reduced basis ${{\color[rgb]{0,0,0}{\Phi}}}$ and substituting the
modal decomposition (11) yields:

${{\color[rgb]{0,0,0}{\Phi}}}^{T}M{{\color[rgb]{0,0,0}{\Phi}}}\dot{\widetilde{\bm{u}}}={{\color[rgb]{0,0,0}{\Phi}}}^{T}\bm{f}(\bar{\bm{u}}+{{\color[rgb]{0,0,0}{\Phi}}}\widetilde{\bm{u}})-{{\color[rgb]{0,0,0}{\Phi}}}^{T}M\dot{\bar{\bm{u}}}.$
(12)

In the present context, the role of the FOM (8) will be played by the coupled
FOM-FOM problem (4). Details of the Galerkin projection for this problem are
given in Sections 5 and 6.

######  Remark 5.

While we focus our attention on the POD method for reduced basis construction
and on the Galerkin method for the projection step, we emphasize that our
approach is not limited to these methods and can be applied to any reduced
order formulation of the coupled FOM-FOM problem that has a provably non-
singular Schur complement. We note that, for nonlinear problems, a third step,
known as hyper-reduction, is needed to treat efficiently the projection of the
nonlinear terms in the governing equations. A variety of approaches for hyper-
reduction exist in the literature, e.g., the Discrete Empirical Method (DEIM)
[50], gappy POD [51], or the Energy Conserving Sampling and Weighting (ECSW)
method [52]. The partitioned solver developed herein is easily extendable to
nonlinear problems, but we omit a detailed discussion of hyper-reduction as
our numerical experiments focus on linear problems. It is straightforward to
see that both the reduced basis construction and Galerkin projection steps of
our model reduction procedure can be precomputed offline, as shown explicitly
later, in Sections 5 and 6.

###  4.2 Reduced basis sets for the transmission problem

We now specialize the first step of the generic POD-based MOR workflow in
Section 4.1 to obtain the reduced basis sets that will be used in this work.
The second step, i.e., the Galerkin projection onto the reduced basis, will be
discussed in Sections 5–6. Let $X_{i}$ denote a set of $r_{i}$ snapshots on
$\Omega_{i}$, $i=1,2$. The columns of $X_{i}$ are the coefficient vectors
$\bm{u}^{m}_{i}$ of finite element solutions $u^{h,m}_{i}\in S^{h}_{i}$,
$m=1,\ldots,r_{i}$. Thus, $X_{i}$ is an $n_{i}\times r_{i}$ matrix. The
columns of $X_{i}$ can be partitioned as
$\bm{u}^{m}_{i}=(\bm{u}^{m}_{i,\gamma},\bm{u}^{m}_{i,0},\bm{g}^{m}_{i})$,
where the coefficient subvectors are defined in Section 3. To handle the
Dirichlet boundary conditions, we adopt an approach similar to the one in
[53]. Specifically, we remove the subvectors $\bm{g}^{m}_{i}$ corresponding to
the Dirichlet nodes to obtain the $n_{i,D}\times r_{i}$ adjusted snapshot
matrix $X_{i,D}$. The $m^{th}$ column of this matrix is given by the vector
$(\bm{u}^{m}_{i,\gamma},\bm{u}^{m}_{i,0})$ for
$\bm{u}^{m}_{i,\gamma}\in\mathbb{R}^{n_{i,\gamma}}$,
$\bm{u}^{m}_{i,0}\in\mathbb{R}^{n_{i,0}}$; see Figure 3.

Figure 3: Left to right: the original snapshot matrix $X_{i}$, the adjusted
snapshot matrix $X_{i,D}$, the interior $X_{i,0}$ and interface $X_{i,\gamma}$
snapshot matrices.

We further split the adjusted snapshot matrix into an $n_{i,0}\times r_{i}$
submatrix $X_{i,0}$ containing all interior nodal values of the snapshots and
an $n_{i,\gamma}\times r_{i}$ submatrix $X_{i,\gamma}$ containing all
interface nodal values of the snapshots. Thus, the columns of $X_{i,0}$ and
$X_{i,\gamma}$ are given by the coefficient vectors $\bm{u}^{m}_{i,0}$ and
$\bm{u}^{m}_{i,\gamma}$, respectively. Figure 3 illustrates the construction
of these companion snapshot matrices.

Next, for $i=1,2$, we apply the POD basis construction to $X_{i,D}$, $X_{i,0}$
and $X_{i,\gamma}$. Specifically, we: (i) compute the SVDs of these matrices,
(ii) choose the integers $0<d_{i,D}\ll n_{i,D}$, $0<d_{i,0}\ll n_{i,0}$ ,and
$0<d_{i,\gamma}\ll n_{i,\gamma}$ that define the percent snapshot energy
captured by the reduced basis for each respective set of snapshots, and (iii)
form the $n_{i,D}\times d_{i,D}$, $n_{i,0}\times d_{i,0}$, and
$n_{i,\gamma}\times d_{i,\gamma}$ reduced bases
${{\color[rgb]{0,0,0}{\Phi}}}_{i,D}$, ${{\color[rgb]{0,0,0}{\Phi}}}_{i,0}$,
and ${{\color[rgb]{0,0,0}{\Phi}}}_{i,\gamma}$, respectively. Because the
columns of ${{\color[rgb]{0,0,0}{\Phi}}}_{i,D}$ contain both the interior and
interface DoFs on $\Omega_{i}$, in the literature they are usually referred to
as the _full subdomain bases_ [13]. We include these bases because they are
ubiquitous in methods that use DD as a vehicle to improve the efficiency of
the MOR workflow; see; e.g., [12, 13, 11]. Similarly, we refer to
${{\color[rgb]{0,0,0}{\Phi}}}_{i,0}$ and
${{\color[rgb]{0,0,0}{\Phi}}}_{i,\gamma}$ as the _interior_ and _interface_
reduced bases, respectively. Once ${{\color[rgb]{0,0,0}{\Phi}}}_{i,D}$,
${{\color[rgb]{0,0,0}{\Phi}}}_{i,0}$, and
${{\color[rgb]{0,0,0}{\Phi}}}_{i,\gamma}$ are obtained, one can approximate
the coefficients of the FOM solution on $\Omega_{i}$, for $i=1,2$, as either

$\bm{u}_{i}(t)\approx\left({{\color[rgb]{0,0,0}{\Phi}}}_{i,D}\widetilde{\bm{u}}_{i,D}(t),\bm{g}_{i}(t)\right)\quad\mbox{or}\quad\bm{u}_{i}(t)\approx\left({{\color[rgb]{0,0,0}{\Phi}}}_{i,\gamma}\widetilde{\bm{u}}_{i,\gamma}(t),{{\color[rgb]{0,0,0}{\Phi}}}_{i,0}\widetilde{\bm{u}}_{i,0}(t),\bm{g}_{i}(t)\right),$
(13)

where $\widetilde{\bm{u}}_{i,D}(t)\in\mathbb{R}^{d_{i,D}}$,
$\widetilde{\bm{u}}_{i,\gamma}(t)\in\mathbb{R}^{d_{i,\gamma}}$, and
$\widetilde{\bm{u}}_{i,0}(t)\in\mathbb{R}^{d_{i,0}}$ are unknown time-
dependent modal amplitudes. It is straightforward to see that both ROM
solutions in (13) will satisfy the prescribed boundary conditions by
construction. In this context, the reference state in (11) is given by
$\bar{\bm{u}}_{i}(t)=(\bm{0},\bm{g}_{i}(t))$, with
$\bm{0}\in\mathbb{R}^{n_{i,D}}$.

##### Reduced bases for the Lagrange multiplier

Because our FOM is given by the coupled problem (4) in which the interface
conditions are enforced by Lagrange multipliers, its Galerkin projection also
requires a suitable reduced basis for the Lagrange multiplier. Such a basis
can be obtained either independently from the RB matrices for the states or
reusing them in a suitable way. In the first case, one collects $r_{\gamma}$
snapshots from some generic Lagrange multiplier space $G^{h}_{\gamma}$ into an
$n_{\gamma}\times r_{\gamma}$ snapshot matrix $X_{\gamma}$ and then follows
the same procedure as above to obtain an $n_{\gamma}\times d_{\gamma}$ reduced
basis matrix ${{\color[rgb]{0,0,0}{\Phi}}}_{\gamma}$. In this paper, we use
solely the second approach and define ${{\color[rgb]{0,0,0}{\Phi}}}_{\gamma}$
using either ${{\color[rgb]{0,0,0}{\Phi}}}_{i,D}$ or
${{\color[rgb]{0,0,0}{\Phi}}}_{i,\gamma}$. The Lagrange multiplier is then
approximated using its reduced basis as

$\bm{\lambda}(t)\approx{{\color[rgb]{0,0,0}{\Phi}}}_{\gamma}\widetilde{\bm{\lambda}}(t),$
(14)

where $\widetilde{\bm{\lambda}}(t)\in\mathbb{R}^{d_{\gamma}}$ is an unknown
time-dependent modal amplitude.

######  Remark 6.

Although the construction of the reduced bases is purely algebraic, their
columns represent coefficient vectors of finite element functions in
$S^{h}_{i}$, for $i=1,2$, and $G^{h}_{\gamma}$. Specifically, using the
columns of ${{\color[rgb]{0,0,0}{\Phi}}}_{i,D}$,
${{\color[rgb]{0,0,0}{\Phi}}}_{i,\gamma}$,
${{\color[rgb]{0,0,0}{\Phi}}}_{i,0}$, and
${{\color[rgb]{0,0,0}{\Phi}}}_{\gamma}$ as coefficients in an expansion in
terms of the nodal basis functions, one obtains functions belonging in
$S^{h}_{i,D}$, $S^{h}_{i,\gamma}$, $S^{h}_{i,0}$, and $G^{h}_{\gamma}$,
respectively. These finite element functions can be viewed as basis sets
spanning _reduced subspaces_ (RS) $\widetilde{S}^{h}_{i,D}$,
$\widetilde{S}^{h}_{i,\gamma}$, $\widetilde{S}^{h}_{i,0}$, and
$\widetilde{G}^{h}_{\gamma}$ of their respective parent finite element spaces.
This functional viewpoint of the reduced bases will be convenient when
analyzing the properties of the partitioned IVR schemes for the coupled ROM-
ROM and ROM-FOM formulations. This analysis is deferred until Section 7.

##  5 An IVR scheme for coupled ROM-ROM problems

In this section, we formulate an IVR scheme for the partitioned solution of
two ROMs coupled across an interface. Formally such a scheme can be obtained
by projecting the coupled FOM-FOM problem (4) onto reduced subspaces
$\widetilde{V}^{h}\subset V^{h}$ and $\widetilde{W}^{h}\subset W^{h}$ and then
using the Schur complement of the resulting coupled ROM-ROM problem to
calculate an accurate approximation of the interface flux.

However, successful execution of this plan requires one to take into
consideration the fact that Galerkin projection of mixed problems does not
automatically preserve their stability properties; see, e.g., [54]. In the
present context this means that the Schur complement of the coupled ROM-ROM
problem is not guaranteed to be non-singular even if the Schur complement of
its parent coupled FOM-FOM has this property.

A sufficient condition for (4) to have a non-singular Schur complement was
established in [1], and requires every Lagrange multiplier to be a trace of a
finite element function from one of the two sides of the interface. In Section
7, we prove that a similar _trace compatibility_ condition ensures that
Galerkin projection of (4) also has this property; see Remark 8. However, this
condition imposes restrictions on the choices of the reduced bases for the
subdomain states and the Lagrange multiplier. In particular, the trace
compatibility condition makes the full subdomain bases
${{\color[rgb]{0,0,0}{\Phi}}}_{i,D}$ (for $i=1,2$) less than an ideal choice
for the extension of the IVR scheme to coupled ROM-ROM problems. To explain
the issues and motivate our approach, let us examine more closely the Galerkin
projection of (4) onto the full subdomain bases. Such a projection uses the
first ansatz in (13) to approximate the subdomain states, i.e.,

$\bm{u}_{i}:=\left({{\color[rgb]{0,0,0}{\Phi}}}_{i,D}\widetilde{\bm{u}}_{i,D},\bm{g}_{i}\right);\quad
i=1,2\,,$ (15)

where $\widetilde{\bm{u}}_{i,D}\in\mathbb{R}^{d_{i,D}}$ are time-dependent
modal amplitudes (reduced order states). One then has to select a reduced
basis (RB) ${{\color[rgb]{0,0,0}{\Phi}}}_{\gamma}$ for the Lagrange multiplier
that is _trace-compatible_ with (15). To construct
${{\color[rgb]{0,0,0}{\Phi}}}_{\gamma}$ note that every column
${\color[rgb]{0,0,0}{\bm{\phi}}}^{j}_{i,D}\in\mathbb{R}^{n_{i,D}}$ of the full
subdomain basis ${{\color[rgb]{0,0,0}{\Phi}}}_{i,D}$ can be partitioned as
${\color[rgb]{0,0,0}{\bm{\phi}}}^{j}_{i,D}=({\color[rgb]{0,0,0}{\bm{\phi}}}^{j}_{i,D,\gamma},{\color[rgb]{0,0,0}{\bm{\phi}}}^{j}_{i,D,0})$,
where
${\color[rgb]{0,0,0}{\bm{\phi}}}^{j}_{i,D,\gamma}\in\mathbb{R}^{n_{i,\gamma}}$
and ${\color[rgb]{0,0,0}{\bm{\phi}}}^{j}_{i,D,0}\in\mathbb{R}^{n_{i,0}}$ are
sub-vectors corresponding to interface and interior degrees of freedom,
respectively. Let ${{\color[rgb]{0,0,0}{\Phi}}}_{i,D,\gamma}$ denote the
$n_{i,\gamma}\times d_{i,D}$ matrix whose $j$th column is given by
${\color[rgb]{0,0,0}{\bm{\phi}}}^{j}_{i,D,\gamma}$. It is easy to see that
${{\color[rgb]{0,0,0}{\Phi}}}_{\gamma}:={{\color[rgb]{0,0,0}{\Phi}}}_{k,D,\gamma}$
for $k=1$ or $k=2$ is a trace-compatible RB for the Lagrange multiplier: every
function in the reduced space $\widetilde{G}^{h}_{\gamma}$ spanned by
${{\color[rgb]{0,0,0}{\Phi}}}_{\gamma}$ is a trace of a function in the
reduced space $\widetilde{S}^{h}_{k,D}$ spanned by
${{\color[rgb]{0,0,0}{\Phi}}}_{k,D}$. However, since sub-vectors of a linearly
independent set of vectors are not necessarily linearly independent on their
own, the reduced order basis ${{\color[rgb]{0,0,0}{\Phi}}}_{i,D,\gamma}$ is
not guaranteed to have a full column rank.

This is a serious drawback for the development of the IVR scheme as it is
easily seen that rank-deficiency of
${{\color[rgb]{0,0,0}{\Phi}}}_{i,D,\gamma}$ will lead to rank-deficiency of
transposed projected constraint matrix
$\widetilde{G}^{T}=(\widetilde{G}_{1,D},-\widetilde{G}_{2,D})^{T}$, thereby
resulting in coupled ROM-ROM problems whose Schur complement is not
invertible. Of course, one can prune the redundant basis functions from
${{\color[rgb]{0,0,0}{\Phi}}}_{i,D,\gamma}$ by computing its SVD and throwing
away all left singular vectors corresponding to zero singular values.
Unfortunately, this solution also suffers from some serious flaws. First, the
“pruned” RB for the Lagrange multiplier may fail to satisfy the original
“snapshot energy” criterion (10) used to select the full subdomain basis
${{\color[rgb]{0,0,0}{\Phi}}}_{i,D}$. Second, to preserve trace compatibility
one would have to project the interface states $\bm{u}_{k,\gamma}$ using the
“pruned” RB, while the interior states $\bm{u}_{k,0}$ will have to be
projected using the RB ${{\color[rgb]{0,0,0}{\Phi}}}_{k,D,0}$ whose columns
are given by the sub-vectors ${\color[rgb]{0,0,0}{\bm{\phi}}}^{j}_{i,D,0}$. It
is clear that ${{\color[rgb]{0,0,0}{\Phi}}}_{k,D,0}$ may suffer from the same
issues as ${{\color[rgb]{0,0,0}{\Phi}}}_{k,D,\gamma}$, that is, it may be
rank-deficient.

Instead of trying to extract a trace-compatible RB for the Lagrange multiplier
from the full subdomain basis and potentially lose its optimality with respect
to the snapshot energy criterion, a more robust strategy is to ensure trace
compatibility from the onset by using separate RBs for the interior and
interface variables. The following section describes the construction of a
coupled ROM-ROM formulation based on this idea.

###  5.1 A coupled ROM-ROM based on a composite reduced basis

Our strategy for securing a coupled ROM-ROM problem with a provably non-
singular Schur complement has two key ingredients. The first one is the
projection of the state $\bm{u}_{i,D}$ using pairs (thus the term “composite”
RB)
$\\{{{\color[rgb]{0,0,0}{\Phi}}}_{i,\gamma},{{\color[rgb]{0,0,0}{\Phi}}}_{i,0}\\}_{i=1,2}$
of _independently_ computed interface and interior RBs instead of a single
full subdomain RB. The second ingredient is achieving trace compatibility by
selecting one of the two interface RBs as a RB for the Lagrange multiplier,
i.e., we set
${{\color[rgb]{0,0,0}{\Phi}}}_{\gamma}={{\color[rgb]{0,0,0}{\Phi}}}_{k,\gamma}$
for $k=1$ or $k=2$. This choice mimics the one in (4) and, as we shall prove
in Section 7, ensures that the Schur complement of the coupled ROM-ROM problem
is non-singular. It also has some similarities with the techniques in [55] and
[13].

To project the coupled FOM-FOM (4) using the composite RB, we apply separate
projections to the interior and to the interface DoFs. Thus, instead of (15),
we use the second ansatz in (13) and set

$\bm{u}_{i}=\left({{\color[rgb]{0,0,0}{\Phi}}}_{i,\gamma}\widetilde{\bm{u}}_{i,\gamma},{{\color[rgb]{0,0,0}{\Phi}}}_{i,0}\widetilde{\bm{u}}_{i,0},\bm{g}_{i}\right)\quad\mbox{and}\quad\bm{\lambda}={{\color[rgb]{0,0,0}{\Phi}}}_{k,\gamma}\widetilde{\bm{\lambda}}.$
(16)

Here, the time-dependent modal amplitudes
$\widetilde{\bm{u}}_{i,\gamma}\in\mathbb{R}^{d_{i,\gamma}}$,
$\widetilde{\bm{u}}_{i,0}\in\mathbb{R}^{d_{i,0}}$, and
$\widetilde{\bm{\lambda}}\in\mathbb{R}^{d_{k,\gamma}}$ represent the reduced
order interface and interior states and the reduced order Lagrange multiplier,
respectively. Following Section 4.1, we insert (16) into (4) and multiply the
blocks corresponding to the interior, interface, and Lagrange multiplier DoFs
by ${{\color[rgb]{0,0,0}{\Phi}}}_{i,0}^{T}$,
${{\color[rgb]{0,0,0}{\Phi}}}_{i,\gamma}^{T}$, and
${{\color[rgb]{0,0,0}{\Phi}}}_{k,\gamma}^{T}$ respectively. These steps yield
the following _composite reduced basis_ coupled ROM-ROM formulation:

$\displaystyle\begin{split}&\begin{bmatrix}\widetilde{M}_{1,\gamma\gamma}&\widetilde{M}_{1,\gamma
0}&0&0&\widetilde{G}_{1,\gamma}^{T}\\\\[4.30554pt]
\widetilde{M}_{1,0\gamma}&\widetilde{M}_{1,00}&0&0&0\\\\[4.30554pt]
0&0&\widetilde{M}_{2,\gamma\gamma}&\widetilde{M}_{2,\gamma
0}&-\widetilde{G}_{2,\gamma}^{T}\\\\[4.30554pt]
0&0&\widetilde{M}_{2,0\gamma}&\widetilde{M}_{2,00}&0\\\\[4.30554pt]
\widetilde{G}_{1,\gamma}&0&-\widetilde{G}_{2,\gamma}&0&0\end{bmatrix}\begin{bmatrix}\dot{\widetilde{\bm{u}}}_{1,\gamma}\\\\[5.38193pt]
\dot{\widetilde{\bm{u}}}_{1,0}\\\\[5.38193pt]
\dot{\widetilde{\bm{u}}}_{2,\gamma}\\\\[5.38193pt]
\dot{\widetilde{\bm{u}}}_{2,0}\\\\[5.38193pt]
\widetilde{\bm{\lambda}}\end{bmatrix}=\begin{bmatrix}\widetilde{\bm{s}}_{1,\gamma}\\\\[5.8125pt]
\widetilde{\bm{s}}_{1,0}\\\\[5.8125pt]
\widetilde{\bm{s}}_{2,\gamma}\\\\[5.8125pt]
\widetilde{\bm{s}}_{2,0}\\\\[5.8125pt]
\widetilde{\bm{s}}_{\gamma}\end{bmatrix}\end{split}\,.$ (17)

The block structure of (16) is induced by the projection onto the composite RB
space. Specifically, we have that, for $i=1,2$,

$\displaystyle\begin{bmatrix}\widetilde{\bm{s}}_{i,\gamma}\\\\[4.30554pt]
\widetilde{\bm{s}}_{i,0}\end{bmatrix}=\begin{bmatrix}\widetilde{\bm{f}}_{i,\gamma}\\\\[4.30554pt]
\widetilde{\bm{f}}_{i,0}\end{bmatrix}-\begin{bmatrix}\widetilde{F}_{i,\gamma\gamma}&\widetilde{F}_{i,\gamma
0}\\\\[2.15277pt]
\widetilde{F}_{i,0\gamma}&\widetilde{F}_{i,00}\end{bmatrix}\begin{bmatrix}\widetilde{\bm{u}}_{i,\gamma}\\\\[4.30554pt]
\widetilde{\bm{u}}_{i,0}\end{bmatrix}-\begin{bmatrix}{{\color[rgb]{0,0,0}{\Phi}}}_{i,\gamma}^{T}Q_{i,\gamma\Gamma}(\dot{\bm{g}}_{i},\bm{g}_{i})\\\\[4.30554pt]
{{\color[rgb]{0,0,0}{\Phi}}}_{i,0}^{T}Q_{i,0\Gamma}(\dot{\bm{g}}_{i},\bm{g}_{i})\end{bmatrix}\,;\quad\widetilde{\bm{s}}_{\gamma}=-{{\color[rgb]{0,0,0}{\Phi}}}^{T}_{k,\gamma}Q_{\gamma,\Gamma}(\dot{\bm{g}}_{1},\dot{\bm{g}}_{2}),$

and that, for $i\in\\{1,2\\}$ and $\\{p,q\\}\in\\{\gamma,0\\}$,

$\begin{array}[]{c}\displaystyle\widetilde{M}_{i,pq}:={{\color[rgb]{0,0,0}{\Phi}}}_{i,p}^{T}M_{i,pq}{{\color[rgb]{0,0,0}{\Phi}}}_{i,q},\quad\widetilde{F}_{i,pq}:={{\color[rgb]{0,0,0}{\Phi}}}_{i,p}^{T}F_{i,pq}{{\color[rgb]{0,0,0}{\Phi}}}_{i,q},\quad\widetilde{G}_{i,\gamma}:={{\color[rgb]{0,0,0}{\Phi}}}_{k,\gamma}^{T}G_{i,\gamma}{{\color[rgb]{0,0,0}{\Phi}}}_{i,\gamma},\\\\[8.61108pt]
\widetilde{\bm{f}}_{i,p}:={{\color[rgb]{0,0,0}{\Phi}}}_{i,p}^{T}\bm{f}_{i,p},\quad
Q_{i,p\Gamma}(\dot{\bm{g}}_{i},\bm{g}_{i})=\
M_{i,p\Gamma}\dot{\bm{g}}_{i}+F_{i,p\Gamma}\bm{g}_{i}.\end{array}$ (18)

For $p\in\\{\gamma,0\\}$, the matrices $M_{i,p\Gamma}$ and $F_{i,p\Gamma}$ are
the blocks of the “partial” mass and flux matrices corresponding to the
interface and interior variables respectively, i.e.,

$M_{i,\Gamma}=[M_{i,\gamma\Gamma};M_{i,0\Gamma}]\quad\mbox{and}\quad
F_{i,\Gamma}=[F_{i,\gamma\Gamma};F_{i,0\Gamma}]\,.$

###  5.2 An IVR scheme for the coupled ROM-ROM based on a composite reduced
basis

In this section, we formulate an IVR scheme for the partitioned solution of
the coupled ROM-ROM problem (17). Analysis in Section 7 provides a rigorous
mathematical basis for this scheme by showing that the Schur complement of
(17) is symmetric and positive definite.

Since this IVR scheme is based on a coupled ROM-ROM problem, similar to other
ROM methods, it comprises an offline stage where one computes the reduced
basis and projects the FOM and an online stage where one uses the resulting
ROM to simulate the system of interest. Although the offline stage of the IVR
ROM-ROM scheme is very similar to that of any POD-based model order reduction
scheme, we include it for completeness of the presentation.

##### Offline: Computation of the composite basis ROMs

  1. 1.

Snapshot collection. Solve the transmission problem (1) using a suitable full
order model to obtain the subdomain snapshot matrices ${X}_{i}$, for $i=1,2$.
Form the adjusted snapshot matrices $X_{i,D}$ and extract their interior
$X_{i,0}$ and interface $X_{i,\gamma}$ parts.

  2. 2.

Reduced basis calculation. For $i=1,2$, choose accuracy thresholds
$\delta_{i,0},\delta_{i,\gamma}>0$, determine the reduced bases dimensions
$d_{i,0}$ and $d_{i,\gamma}$ as in (10), and calculate the reduced bases
${{\color[rgb]{0,0,0}{\Phi}}}_{i,0}$,
${{\color[rgb]{0,0,0}{\Phi}}}_{i,\gamma}$ following Section 4.2. Choose $k=1$
or $k=2$ and set
${{\color[rgb]{0,0,0}{\Phi}}}_{\gamma}={{\color[rgb]{0,0,0}{\Phi}}}_{k,\gamma}$.

  3. 3.

Galerkin projection. For $i=1,2$ and $\\{p,q\\}\in\\{0,\gamma\\}$, precompute
the ROM matrices:

$\begin{array}[]{c}\widetilde{M}_{i,pq}:={{\color[rgb]{0,0,0}{\Phi}}}^{T}_{i,p}M_{i,pq}{{\color[rgb]{0,0,0}{\Phi}}}_{i,q}\in\mathbb{R}^{d_{i,p}\times
d_{i,q}};\ \
\widetilde{F}_{i,pq}:={{\color[rgb]{0,0,0}{\Phi}}}_{i,p}^{T}F_{i,pq}{{\color[rgb]{0,0,0}{\Phi}}}_{i,q}\in\mathbb{R}^{d_{i,p}\times
d_{i,q}};\ \
\widetilde{G}_{i}:={{\color[rgb]{0,0,0}{\Phi}}}_{k,\gamma}^{T}G_{i}{{\color[rgb]{0,0,0}{\Phi}}}_{i,\gamma}\in\mathbb{R}^{d_{k,\gamma}\times
d_{i,\gamma}}\\\\[8.61108pt]
\widetilde{Q}_{i,p\Gamma}=\\{{{\color[rgb]{0,0,0}{\Phi}}}_{i,p}^{T}M_{i,p\Gamma},{{\color[rgb]{0,0,0}{\Phi}}}_{i,p}^{T}F_{i,p\Gamma}\\},\quad\mbox{and}\quad\widetilde{Q}_{\gamma,\Gamma}=\\{{{\color[rgb]{0,0,0}{\Phi}}}^{T}_{k,\gamma}G_{1,\Gamma},-{{\color[rgb]{0,0,0}{\Phi}}}^{T}_{k,\gamma}G_{2,\Gamma}\\}\end{array}\,.$

##### Online: Partitioned solution of the coupled ROM-ROM system (17)

  1. 1.

Given a simulation time interval $[0,T]$ choose explicit time integration
schemes $D^{n}_{i,t}(\widetilde{\bm{u}})$ on $\Omega_{i}$, $i=1,2$.

  2. 2.

For $i=1,2$, $p\in\\{0,\gamma\\}$ and $n=0,1,\ldots$, compute the right-hand
side vectors
$\widetilde{\bm{f}}^{n}_{i,p}:={{\color[rgb]{0,0,0}{\Phi}}}_{i,p}^{T}\bm{f}^{n}_{i,p}$,

$\displaystyle\begin{bmatrix}\widetilde{\bm{s}}^{n}_{i,\gamma}\\\\[4.30554pt]
\widetilde{\bm{s}}^{n}_{i,0}\end{bmatrix}=\begin{bmatrix}\widetilde{\bm{f}}^{n}_{i,\gamma}\\\\[4.30554pt]
\widetilde{\bm{f}}^{n}_{i,0}\end{bmatrix}-\begin{bmatrix}\widetilde{F}_{i,\gamma\gamma}&\widetilde{F}_{i,\gamma
0}\\\\[2.15277pt]
\widetilde{F}_{i,0\gamma}&\widetilde{F}_{i,00}\end{bmatrix}\begin{bmatrix}\widetilde{\bm{u}}^{n}_{i,\gamma}\\\\[4.30554pt]
\widetilde{\bm{u}}^{n}_{i,0}\end{bmatrix}-\begin{bmatrix}\widetilde{Q}_{i,\gamma\Gamma}(\dot{\bm{g}}^{n}_{i},\bm{g}^{n}_{i})\\\\[4.30554pt]
\widetilde{Q}_{i,0\Gamma}(\dot{\bm{g}}^{n}_{i},\bm{g}^{n}_{i})\end{bmatrix}\,,\quad\mbox{and}\quad\widetilde{\bm{s}}^{n}_{\gamma}=-\widetilde{Q}_{\gamma,\Gamma}(\dot{\bm{g}}^{n}_{1},\dot{\bm{g}}^{n}_{2}),$

where $\dot{\bm{g}}^{n}_{i}$ is approximation of the time derivative of the
boundary data at the current time step.

  3. 3.

For $i=1,2$, let $\widetilde{M}_{i}$, $\widetilde{G}_{i}$,
$\widetilde{\bm{s}}^{n}_{i}$ and $\widetilde{\bm{u}}^{n}_{i}$ denote the
$2\times 2$, and $2\times 1$ block matrices and vectors in (17). Solve the
Schur complement system for $\widetilde{\bm{\lambda}}^{n}$:

$\Big{(}\widetilde{G}_{1}\widetilde{M}_{1}^{-1}\widetilde{G}_{1}^{T}+\widetilde{G}_{2}\widetilde{M}_{2}^{-1}\widetilde{G}_{2}^{T}\Big{)}\widetilde{\bm{\lambda}}^{n}=\widetilde{G}_{1}\widetilde{M}_{1}^{-1}\widetilde{\bm{s}}_{1}^{n}-\widetilde{G}_{2}\widetilde{M}_{2}^{-1}\widetilde{\bm{s}}_{2}^{n}-\widetilde{\bm{s}}_{\gamma}^{n}\,.$
(19)

  4. 4.

For $i=1,2$, solve the subdomain ROM problems

$\begin{bmatrix}\widetilde{M}_{i,\gamma\gamma}&\widetilde{M}_{i,\gamma
0}\\\\[4.30554pt]
\widetilde{M}_{i,0\gamma}&\widetilde{M}_{i,00}\end{bmatrix}D^{n}_{i,t}\left(\begin{bmatrix}\widetilde{\bm{u}}_{i,\gamma}^{n+1}\\\\[4.30554pt]
\widetilde{\bm{u}}_{i,0}^{n+1}\end{bmatrix}\right)=\begin{bmatrix}\widetilde{\bm{s}}_{i,\gamma}^{n}+(-1)^{i}\widetilde{G}_{i}^{T}\widetilde{\bm{\lambda}}^{n}\\\\[4.30554pt]
\widetilde{\bm{s}}_{i,0}^{n}\end{bmatrix}$

for the ROM solution $\widetilde{\bm{u}}_{i}^{n+1}$ at the new time step.

  5. 5.

For $i=1,2$, project the ROM solutions to the state spaces of the FOMs on
$\Omega_{i}$:

$\bm{u}^{n+1}_{i}=\left({{\color[rgb]{0,0,0}{\Phi}}}_{i,\gamma}\widetilde{\bm{u}}^{n+1}_{i,\gamma},{{\color[rgb]{0,0,0}{\Phi}}}_{i,0}\widetilde{\bm{u}}^{n+1}_{i,0},\bm{g}_{i}^{n+1}\right)\,.$

##  6 An IVR scheme for coupled ROM-FOM problems

This section extends the IVR scheme to the case of a ROM coupled with a FOM.
Such a formulation is relevant to multiple simulation scenarios for the model
transmission problem (1). One such scenario is when one of the subdomains is
much larger than the other and its FOM dominates the computational cost. To
balance the computational costs across the subdomains, one can replace the FOM
on the large domain with a computationally efficient ROM, while retaining the
FOM on the small subdomain. A second possible scenario is when the governing
equations are parameterized on just one of the subdomains and simulation of
the other subdomain requires a scheme that can handle all admissible inputs.
In this case, a ROM would be only appropriate for the domain with the
parameterized equations, while on the other subdomain one would still have to
use a FOM. A related scenario is the case where a coupled ROM-FOM model has
the potential of having better predictive accuracy than a model based solely
on ROMs [9, 12, 18, 56, 57]. One example of this scenario would be a version
of our model transmission problem (1) in which the diffusion coefficient on
one of the subdomains is allowed to be identically zero. In this case, the
governing equation on that subdomain reduces to a pure advection problem for
which POD-based MOR is not effective; see Section 4.

###  6.1 A coupled ROM-FOM based on a composite reduced basis

As in the coupled ROM-ROM case, to develop an IVR scheme for the partitioned
solution of coupled ROM-FOM problems, we first formulate a coupled ROM-FOM
problem that is guaranteed to have a symmetric and positive definite Schur
complement. To obtain this problem, assume that the FOM should be retained on
$\Omega_{2}$ while simulation on $\Omega_{1}$ can be performed by a ROM. To
define the corresponding coupled ROM-FOM, we start from the coupled FOM-FOM
(4), retain the FOM on $\Omega_{2}$, and project the state on $\Omega_{1}$
${{\color[rgb]{0,0,0}{\Phi}}}_{1,C}:=\\{{{\color[rgb]{0,0,0}{\Phi}}}_{1,\gamma},{{\color[rgb]{0,0,0}{\Phi}}}_{1,0}\\}$
using the composite RB ansatz

$\bm{u}_{1}=\left({{\color[rgb]{0,0,0}{\Phi}}}_{1,\gamma}\widetilde{\bm{u}}_{1,\gamma},{{\color[rgb]{0,0,0}{\Phi}}}_{1,0}\widetilde{\bm{u}}_{1,0},\bm{g}_{1}\right)\,.$
(20)

To complete the coupled ROM-FOM problem, one has to choose a trace-compatible
representation for the Lagrange multiplier. In the present setting there are
two possible options that satisfy this condition: the interface RB
${{\color[rgb]{0,0,0}{\Phi}}}_{1,\gamma}$ from the ROM side of the interface
(option “rLM”) or the interface finite element space $G^{h}_{2}$ from the FOM
side of the interface (option “fLM”). In the first case
$\bm{\lambda}={{\color[rgb]{0,0,0}{\Phi}}}_{1,\gamma}\widetilde{\bm{\lambda}}$
and in the second case $\bm{\lambda}$ is the coefficient vector of a function
$\lambda^{h}\in G^{h}_{2}$. Analysis in Section 7 will confirm that either one
of these two options leads to coupled problems with non-singular Schur
complements.

With these choices, we have the following _composite RB_ coupled ROM-FOM
formulation:

$\displaystyle\begin{split}&\begin{bmatrix}\widetilde{M}_{1,\gamma\gamma}&\widetilde{M}_{1,\gamma
0}&0&0&\widehat{G}_{1,\gamma}^{T}\\\\[4.30554pt]
\widetilde{M}_{1,0\gamma}&\widetilde{M}_{1,00}&0&0&0\\\\[4.30554pt]
0&0&{M}_{2,\gamma\gamma}&{M}_{2,\gamma
0}&-\widehat{G}_{2,\gamma}^{T}\\\\[4.30554pt]
0&0&{M}_{2,0\gamma}&{M}_{2,00}&0\\\\[4.30554pt]
\widehat{G}_{1,\gamma}&0&-\widehat{G}_{2,\gamma}&0&0\end{bmatrix}\begin{bmatrix}\dot{\widetilde{\bm{u}}}_{1,\gamma}\\\\[5.38193pt]
\dot{\widetilde{\bm{u}}}_{1,0}\\\\[5.38193pt]
\dot{\bm{u}}_{2,\gamma}\\\\[5.38193pt] \dot{\bm{u}}_{2,0}\\\\[5.38193pt]
\widehat{\bm{\lambda}}\end{bmatrix}=\begin{bmatrix}\widetilde{\bm{s}}_{1,\gamma}\\\\[5.8125pt]
\widetilde{\bm{s}}_{1,0}\\\\[5.8125pt] \bm{s}_{2,\gamma}\\\\[5.8125pt]
\bm{s}_{2,0}\\\\[5.8125pt]
\widehat{\bm{s}}_{\gamma}\end{bmatrix}\end{split}\,,$ (21)

where the blocks with the “tilde” accent are defined as in the coupled ROM-ROM
(16), the blocks without accents are defined as in the coupled FOM-FOM (4),
and

$\begin{array}[]{c}\widehat{\bm{\lambda}}=\left\\{\\!\\!\begin{array}[]{rl}\widetilde{\bm{\lambda}},\\!&\\!\mbox{for
option rLM}\\\\[4.30554pt] \bm{\lambda},\\!&\\!\mbox{for option
fLM}\end{array}\,;\right.\widehat{\bm{s}}_{\gamma}=\left\\{\\!\\!\begin{array}[]{rl}-{{\color[rgb]{0,0,0}{\Phi}}}^{T}_{1,\gamma}Q_{\gamma,\Gamma}(\dot{\bm{g}}_{1},\dot{\bm{g}}_{2}),\\!&\\!\mbox{for
option rLM}\\\\[4.30554pt]
-Q_{\gamma,\Gamma}(\dot{\bm{g}}_{1},\dot{\bm{g}}_{2}),\\!&\\!\mbox{for option
fLM}\end{array}\right.;\\\ \\\
\widehat{G}_{1,\gamma}=\left\\{\\!\\!\begin{array}[]{rl}{{\color[rgb]{0,0,0}{\Phi}}}^{T}_{1,\gamma}G_{1,\gamma}{{\color[rgb]{0,0,0}{\Phi}}}_{1,\gamma},\\!&\\!\mbox{for
option rLM}\\\\[4.30554pt]
G_{1,\gamma}{{\color[rgb]{0,0,0}{\Phi}}}_{1,\gamma},\\!&\\!\mbox{for option
fLM}\end{array}\,\right.;\;\widehat{G}_{2,\gamma}=\left\\{\\!\\!\begin{array}[]{rl}{{\color[rgb]{0,0,0}{\Phi}}}^{T}_{1,\gamma}G_{2,\gamma},\\!&\\!\mbox{for
option rLM}\\\\[4.30554pt] G_{2,\gamma},\\!&\\!\mbox{for option
fLM}\end{array}\,\right..\end{array}$ (22)

In the next section, we present the IVR scheme for the partitioned solution of
(21).

###  6.2 An IVR scheme for the coupled ROM-FOM based on a composite reduced
basis

Since the coupled ROM-FOM problem has a ROM component, the IVR scheme for (21)
necessarily involves an offline stage where one computes the reduced basis and
projects the FOM on $\Omega_{1}$ and an online stage where one uses the
coupled problem to simulate the model transmission problem. Although these
stages are very similar to those outlined in Section 5.2, we include an
abridged version for the convenience of the reader. To reduce notational
clutter in some cases we switch to the more compact notation $M_{2,D}$,
$\bm{s}_{2,D}$, and $\bm{f}_{2,D}$ for the matrix and vector blocks of the FOM
problem.

##### Offline: Computation of the composite basis ROM for $\Omega_{1}$

  1. 1.

Snapshot collection. Solve the transmission problem (1) using a suitable full
order model to obtain the snapshot matrix $X_{1}$, form $X_{1,D}$ and extract
$X_{1,\gamma}$ and $X_{1,0}$.

  2. 2.

Reduced basis calculation. Choose accuracy thresholds
$\delta_{1,0},\delta_{1,\gamma}>0$, determine the reduced bases dimensions
$d_{1,0}$ and $d_{1\gamma}$, and calculate the reduced bases
${{\color[rgb]{0,0,0}{\Phi}}}_{1,0}$,
${{\color[rgb]{0,0,0}{\Phi}}}_{1,\gamma}$. Select an option (rLM or fLM) for
the Lagrange multiplier.

  3. 3.

Galerkin projection. For $\\{p,q\\}\in\\{0,\gamma\\}$, precompute the ROM
matrices $\widetilde{M}_{1,pq}$, $\widetilde{F}_{1,pq}$,
$\widetilde{Q}_{1,p\Gamma}$. Precompute $\widehat{G}_{i,\gamma}$ as defined in
(22) and

$\widehat{Q}_{\gamma,\Gamma}:=\left\\{\begin{array}[]{rl}\\{{{\color[rgb]{0,0,0}{\Phi}}}^{T}_{1,\gamma}G_{1,\Gamma},-{{\color[rgb]{0,0,0}{\Phi}}}^{T}_{1,\gamma}G_{2,\Gamma}\\},&\mbox{for
option rLM}\\\\[4.30554pt] \\{G_{1,\Gamma},-G_{2,\Gamma}\\},&\mbox{for option
fLM}\end{array}\right..$

##### Online: Partitioned solution of the composite basis coupled ROM-FOM
system

  1. 1.

Given a simulation time interval $[0,T]$ choose explicit time integration
schemes $D^{n}_{i,t}(\widetilde{\bm{u}})$ on $\Omega_{i}$, $i=1,2$.

  2. 2.

For $p\in\\{0,\gamma\\}$ and and $n=0,1,\ldots$, compute the ROM vectors
$\widetilde{\bm{f}}^{n}_{1,p}:={{\color[rgb]{0,0,0}{\Phi}}}_{1,p}^{T}\bm{f}^{n}_{1,p}$,
$\widetilde{\bm{s}}^{n}_{1,\gamma}$, $\widetilde{\bm{s}}^{n}_{1,0}$ as in
Section 5.2. Compute the FOM vectors $\bm{f}^{n}_{2,D}$, ${\bm{s}}^{n}_{2,D}$,
and the right-hand side for the constraint equation:

$\widehat{\bm{s}}^{n}_{\gamma}=-\widehat{Q}_{\gamma,\Gamma}(\dot{\bm{g}}^{n}_{1},\dot{\bm{g}}^{n}_{2}).$

  3. 3.

Solve the Schur complement system for $\widehat{\bm{\lambda}}^{n}$:

$\Big{(}\widehat{G}_{1}\widetilde{M}_{1}^{-1}\widehat{G}_{1}^{T}+\widehat{G}_{2}{M}_{2,D}^{-1}\widehat{G}_{2}^{T}\Big{)}\widehat{\bm{\lambda}}^{n}=\widehat{G}_{1}\widetilde{M}_{1}^{-1}\widetilde{\bm{s}}_{1}^{n}-\widehat{G}_{2}{M}_{2,D}^{-1}{\bm{s}}_{2}^{n}-\widehat{\bm{s}}_{\gamma}^{n}\,.$
(23)

  4. 4.

Solve the subdomain ROM problem

$\begin{bmatrix}\widetilde{M}_{1,\gamma\gamma}&\widetilde{M}_{1,\gamma 0}\\\
\widetilde{M}_{1,0\gamma}&\widetilde{M}_{1,00}\end{bmatrix}D^{n}_{1,t}\left(\begin{bmatrix}\widetilde{\bm{u}}_{1,\gamma}^{n+1}\\\
\widetilde{\bm{u}}_{1,0}^{n+1}\end{bmatrix}\right)=\begin{bmatrix}\widetilde{\bm{s}}_{1,\gamma}^{n}-\widehat{G}_{1}^{T}\widehat{\bm{\lambda}}^{n}\\\
\widetilde{\bm{s}}_{i,0}^{n}\end{bmatrix},$

to obtain the ROM solution $\widetilde{\bm{u}}_{1}^{n+1}$ at the new time
step. Solve the subdomain FOM problem

$\begin{bmatrix}{M}_{2,\gamma\gamma}&{M}_{2,\gamma 0}\\\
{M}_{2,0\gamma}&{M}_{2,00}\end{bmatrix}D^{n}_{2,t}\left(\begin{bmatrix}{\bm{u}}_{2,\gamma}^{n+1}\\\
{\bm{u}}_{2,0}^{n+1}\end{bmatrix}\right)=\begin{bmatrix}{\bm{s}}_{2,\gamma}^{n}+\widehat{G}_{2}^{T}\widehat{\bm{\lambda}}^{n}\\\\[4.30554pt]
{\bm{s}}_{2,0}^{n}\end{bmatrix},$

to obtain the FOM solution ${\bm{u}}_{2}^{n+1}$ at the new time step.

  5. 5.

Project the ROM solution to the state space of the FOM on $\Omega_{1}$:

$\bm{u}^{n+1}_{1}=\left({{\color[rgb]{0,0,0}{\Phi}}}_{1,\gamma}\widetilde{\bm{u}}^{n+1}_{1,\gamma},{{\color[rgb]{0,0,0}{\Phi}}}_{1,0}\widetilde{\bm{u}}^{n+1}_{1,0},\bm{g}_{1}^{n+1}\right)\,.$

In the next section, we show that both the coupled ROM-ROM and ROM-FOM have
provably non-singular Schur complements, thereby providing appropriate
settings for an application of the IVR scheme.

##  7 Analysis

We will first consider the coupled ROM-ROM formulation (17) and use
variational techniques to prove that it has a symmetric and positive definite
Schur complement. We will then specialize this analysis to the case of the
coupled ROM-FOM problem.

###  7.1 Composite reduced basis coupled ROM-ROM

Successful completion of the online stage of the IVR scheme for (17) hinges on
the unique solvability of the Schur complement system (19) in Step 3. We will
show that this system is uniquely solvable by proving that the Schur
complement matrix

$\widetilde{S}:=\widetilde{G}_{1}\widetilde{M}_{1}^{-1}\widetilde{G}_{1}^{T}+\widetilde{G}_{2}\widetilde{M}_{2}^{-1}\widetilde{G}_{2}^{T}$
(24)

is symmetric and positive definite (SPD). It is well-known that this property
requires the projected mass matrices $\widetilde{M}_{i}$, for $i=1,2$, to be
symmetric and positive definite and the matrix
$\widetilde{G}^{T}=(\widetilde{G}_{1},-\widetilde{G}_{2})^{T}$ to have full
column rank.

While it is possible to develop purely algebraic proofs of the required
properties, here we adopt a variational approach that exploits connections
between the matrix defining the left hand side of the coupled ROM-ROM problem
and mixed variational forms. In so doing, we illuminate how properties of the
variational formulation underlying (17) translate into properties of its
algebraic equivalent. This approach can also expose potential dependencies of
these properties on the dimension of the reduced basis and/or the mesh
parameter. The latter is not easily achievable through strictly algebraic
means.

Let $V^{h}_{D}=S^{h}_{1,D}\times S^{h}_{2,D}$ and $W^{h}=G^{h}_{k}$. We
introduce the _auxiliary_ mixed variational form $B:(V^{h}_{D}\times
W^{h})\times(V^{h}_{D}\times W^{h})\mapsto\mathbb{R}$ as

$B({u}^{h}_{1,D},{u}^{h}_{2,D},\mu^{h};{v}^{h}_{1,D},{v}^{h}_{2,D};\lambda^{h}):=a({u}^{h}_{1,D},{u}^{h}_{2,D};{v}^{h}_{1,D},{v}^{h}_{2,D})+b({v}^{h}_{1,D},{v}^{h}_{2,D};\lambda^{h})+b({u}^{h}_{1,D},{u}^{h}_{2,D};\mu^{h})$
(25)

where $a:V^{h}_{D}\times V^{h}_{D}\mapsto\mathbb{R}$ and $b:V^{h}_{D}\times
W^{h}\mapsto\mathbb{R}$ are defined as

$a({u}^{h}_{1,D},{u}^{h}_{2,D};{v}^{h}_{1,D},{v}^{h}_{2,D})=\left(u^{h}_{1,D},v^{h}_{1,D}\right)_{0,\Omega_{1}}+\left(u^{h}_{2,D},v^{h}_{2,D}\right)_{0,\Omega_{2}}\
\ \mbox{and}\ \
b({v}^{h}_{1,D},{v}^{h}_{2,D};\lambda^{h})=\left({v}^{h}_{1,D}-{v}^{h}_{2,D},\lambda^{h}\right)_{0,\gamma},$

respectively. We call the bilinear form (25) “auxiliary” because it is not the
form that corresponds to the weak coupled problem (3); rather, it is the form
that generates the matrix operators on the left-hand side of the coupled FOM-
FOM problem (4).

To prove that the Schur complement (24) is SPD, we will apply Brezzi’s mixed
variational theory [43] to (25) to show that the projected mass matrices are
SPD and that the transpose constraint matrix has full column rank. Application
of this mixed theory requires a proper functional setting for (25),
specifically the endowment of $V^{h}_{D}$ and $W^{h}$ with suitable norms. We
define these norms as

$\|\\{{v}^{h}_{1,D},{v}^{h}_{2,D}\\}\|^{2}_{V}:=\|{v}^{h}_{1,D}\|^{2}_{0,\Omega_{1}}+\|{v}^{h}_{2,D}\|^{2}_{0,\Omega_{2}}\quad\mbox{and}\quad\|\lambda^{h}\|_{W}:=\|\lambda_{h}\|_{0,\gamma},$
(26)

respectively.

######  Remark 7.

Although the IVR scheme bears resemblance with DD methods based on Lagrange
multipliers such as FETI [58, 59] and mortar methods [45], the variational
setting for its analysis provided by (25) and (26) is different from that
required for the analysis of these DD schemes. This difference stems from the
fact that analysis of IVR relies on the auxiliary form (25), which does not
include any contributions from the flux terms, whereas analysis of DD methods
involves the “true” mixed form corresponding to (3), which includes such
terms. A proper functional setting for the latter requires a broken $H^{1}$
norm on $V^{h}_{D}$, instead of the broken $L^{2}$ norm (26) used here, and a
discrete approximation of the trace norm on $H^{-1/2}(\gamma)$. The use of the
auxiliary form (25) relaxes the requirements on the functional setting for the
application of the mixed theory and allows us to use a weaker norm on
$V^{h}_{D}$ and a very “crude” approximation of the trace norm by an $L^{2}$
norm on $\gamma$. We refer to [60] and [61] for further details about the
analysis of DD methods.

To apply the mixed theory to (17), we need to further adjust the variational
setting so that the auxiliary form (25) produces the left-hand side of the
coupled ROM-ROM problem. For $i=1,2$, let $\widetilde{S}^{h}_{i,\gamma}$,
$\widetilde{S}^{h}_{i,0}$, and $\widetilde{G}^{h}_{k}$ be the reduced
subspaces of ${S}^{h}_{i,\gamma}$, ${S}^{h}_{i,0}$, and ${G}^{h}_{k}$, induced
by the columns of the composite RBs ${\color[rgb]{0,0,0}{\Phi}}_{i,\gamma}$,
${\color[rgb]{0,0,0}{\Phi}}_{i,0}$, and
${\color[rgb]{0,0,0}{\Phi}}_{k,\gamma}$, respectively; see Section 4. We
define the reduced subspaces $\widetilde{V}^{h}_{C}\subset V^{h}_{D}$ and
$\widetilde{W}^{h}\subset W^{h}$ as

$\widetilde{V}^{h}_{C}=\widetilde{S}^{h}_{1,C}\times\widetilde{S}^{h}_{2,C}\quad\mbox{and}\quad\widetilde{W}^{h}=\widetilde{G}^{h}_{k}\,,$
(27)

respectively, where555Note that we have deviated from our usual naming
convention and have labeled the subspace of $S^{h}_{i,D}$ engendered by the
composite basis as $\widetilde{S}^{h}_{i,C}$. This is done in order to avoid
confusion with the reduced subspace $\widetilde{S}^{h}_{i,D}$, which
corresponds to the full subdomain RB matrix
${\color[rgb]{0,0,0}{\Phi}}_{i,D}$.
$\widetilde{S}^{h}_{i,C}:=\widetilde{S}^{h}_{i,\gamma}\oplus\widetilde{S}^{h}_{i,0}$.

It is straightforward to check that restriction of the auxiliary form (25) to
the reduced subspaces (27) generates the matrix on the left-hand side of the
coupled ROM-ROM problem (17). For example, consider a finite element function
${u}^{h}_{i,C}\in\widetilde{S}^{h}_{i,C}$ with coefficient vector
$\bm{u}_{i,C}=(\bm{u}_{i,\gamma},\bm{u}_{i,0})$, where

$\bm{u}_{i,\gamma}={\color[rgb]{0,0,0}{\Phi}}_{i,\gamma}\widetilde{\bm{u}}_{i,\gamma}\quad\mbox{and}\quad\bm{u}_{i,0}={\color[rgb]{0,0,0}{\Phi}}_{i,0}\widetilde{\bm{u}}_{i,0},$
(28)

for some modal amplitudes $\widetilde{\bm{u}}_{i,\gamma}$ and
$\widetilde{\bm{u}}_{i,0}$. Let now
$\\{{u}^{h}_{1,C},{u}^{h}_{2,C}\\},\\{{v}^{h}_{1,C},{v}^{h}_{2,C}\\}\in\widetilde{V}^{h}_{C}$.
Using (28) it easily follows that

$a({u}^{h}_{1,C},{u}^{h}_{2,C};{v}^{h}_{1,C},{v}^{h}_{2,C})=\sum_{i=1}^{2}\begin{bmatrix}\widetilde{\bm{u}}^{T}_{i,\gamma}&\widetilde{\bm{u}}^{T}_{i,0}\end{bmatrix}\begin{bmatrix}\widetilde{M}_{i,\gamma\gamma}&\widetilde{M}_{i,\gamma
0}\\\
\widetilde{M}_{i,0\gamma}&\widetilde{M}_{i,00}\end{bmatrix}\begin{bmatrix}\widetilde{\bm{u}}_{i,\gamma}\\\
\widetilde{\bm{u}}_{i,0}\,\end{bmatrix}.$ (29)

Application of the Brezzi theory requires verification of two separate
conditions on $a(\cdot,\cdot)$ and $b(\cdot,\cdot)$. Specialized to the
functional setting for (17) constructed earlier, these conditions are as
follows.

##### Coercivity on the kernel

The form $a(\cdot,\cdot)$ is coercive on the nullspace
$\widetilde{Z}^{h}_{C}\subset\widetilde{V}^{h}_{C}$ of $b(\cdot,\cdot)$
defined as

$\widetilde{Z}^{h}_{C}=\left\\{\\{{v}^{h}_{1,C},{v}^{h}_{2,C}\\}\in\widetilde{V}^{h}_{C}\,\big{|}b({v}^{h}_{1,C},{v}^{h}_{2,C};\mu^{h})=0\quad\forall\mu^{h}\in\widetilde{W}^{h}\right\\}\,.$

##### Inf-sup condition

For any $\mu^{h}\in\widetilde{W}^{h}$ the form $b(\cdot,\cdot)$ satisfies the
inequality

$\sup_{\\{{v}^{h}_{1,C},{v}^{h}_{2,C}\\}\in\widetilde{V}^{h}_{C}\times\widetilde{V}^{h}_{C}}\frac{b({v}^{h}_{1,C},{v}^{h}_{2,C};{\mu}^{h})}{\|\\{{v}^{h}_{1,C},{v}^{h}_{2,C}\\}\|_{V}}\geq\beta\|\mu^{h}\|_{W},$
(30)

with a mesh-independent constant $\beta$.

The proof of the first Brezzi condition is trivial as it is easy to see that
$a(\cdot,\cdot)$ is coercive on all of $V^{h}_{D}\times V^{h}_{D}$. Since
strong coercivity is inherited on subspaces, it follows that $a(\cdot,\cdot)$
is coercive on $\widetilde{V}^{h}_{C}\times\widetilde{V}^{h}_{C}$ and its
subspace $\widetilde{Z}^{h}_{C}\times\widetilde{Z}^{h}_{C}$. Using (29), it
easily follows that the algebraic translation of this property amounts to the
statement that the projected mass matrices are SPD, which verifies the first
condition necessary to establish that the Schur complement (24) is SPD.

It is well-known that the second condition, i.e., the requirement that
$\widetilde{G}^{T}$ has full column rank, is a consequence of $b(\cdot,\cdot)$
satisfying the inf-sup condition; see, e.g., [62, p.38, Proposition 3.1] for a
discussion of the relationships between algebraic and variational properties
of discrete mixed problems. To prove the inf-sup condition, we will need the
following auxiliary result.

######  Lemma 1.

Let $h_{k}$ denote the characteristic element size of the interface mesh
defining the Lagrange multiplier space $G^{h}_{k}$. There exists an operator
$\mathcal{Q}:\widetilde{W}^{h}\mapsto\widetilde{V}^{h}_{C}$ such that for
every $\mu^{h}\in\widetilde{W}^{h}$ there holds

$(a)\ \ \|\mu^{h}\|^{2}_{W}\leq
C_{1}b(\mathcal{Q}(\mu^{h});\mu^{h})\quad\mbox{and}\quad(b)\ \
\|\mathcal{Q}(\mu^{h})\|_{V}\leq C_{2}h^{\alpha}_{k}\|\mu^{h}\|_{W}\,,$ (31)

where $\alpha\geq 0$ and $C_{1}$, $C_{2}$ are positive constants independent
of this element size.

###### Proof.

Let $k=1$ or $k=2$ be the index used to define the reduced basis for the
Lagrange multiplier space. According to (28), the coefficient vector of a
function ${u}^{h}_{k,C}\in\widetilde{S}^{h}_{k,C}$ is given by
$\bm{u}_{k,C}=({\color[rgb]{0,0,0}{\Phi}}_{k,\gamma}\widetilde{\bm{u}}_{k,\gamma},{\color[rgb]{0,0,0}{\Phi}}_{k,0}\widetilde{\bm{u}}_{k,0})$,
where $\widetilde{\bm{u}}_{k,\gamma}\in\mathbb{R}^{d_{k,\gamma}}$ and
$\widetilde{\bm{u}}_{k,0}\in\mathbb{R}^{d_{k,0}}$ are the associated interface
and interior modal amplitudes. Let $\mu^{h}\in\widetilde{W}^{h}$ be an
arbitrary function in the reduced Lagrange multiplier space. The coefficient
vector of this function is given by the ansatz in (16), i.e.,
$\bm{\mu}={\color[rgb]{0,0,0}{\Phi}}_{k,\gamma}\widetilde{\bm{\mu}}$, with a
modal amplitude $\widetilde{\bm{\mu}}\in\mathbb{R}^{d_{k,\gamma}}$. It follows
that the coefficient vector

$\bm{u}^{\mu}_{k,C}=(\bm{\mu},\bm{0})=({\color[rgb]{0,0,0}{\Phi}}_{k,\gamma}\widetilde{\bm{\mu}},\bm{0});\quad\bm{0}\in\mathbb{R}^{n_{k,0}}$
(32)

defines a lifting ${u}^{h,\mu}_{k,C}\in\widetilde{S}^{h}_{k,C}$ of $\mu^{h}$
such that

${u}^{h,\mu}_{k,C}\big{|}_{\gamma}=\mu^{h}\,.$

We define the operator $\mathcal{Q}=\\{\mathcal{Q}_{1},\mathcal{Q}_{2}\\}$
using this lifting as

$\mathcal{Q}(\mu^{h})=\left\\{\begin{array}[]{rl}\\{{u}^{h,\mu}_{1,C},0\\}\in\widetilde{V}^{h}_{C}&\mbox{if
$k=1$}\\\\[6.45831pt]
-\\{0,{u}^{h,\mu}_{2,C}\\}\in\widetilde{V}^{h}_{C}&\mbox{if
$k=2$}\end{array}\right.\,.$ (33)

With this definition the first assertion in (31) holds trivially with
$C_{1}=1$:

$b(\mathcal{Q}(\mu^{h});\mu^{h})=\left(\mathcal{Q}_{1}(\mu^{h})-\mathcal{Q}_{2}(\mu^{h}),\mu^{h}\right)_{0,\gamma}=\left(\mu^{h},\mu^{h}\right)_{0,\gamma}=\|\mu^{h}\|^{2}_{W}\,.$

To prove the second assertion in (31), we start by noting that

$\|\mathcal{Q}(\mu^{h})\|^{2}_{V}=\|\mathcal{Q}_{1}(\mu^{h})\|^{2}_{0,\Omega_{1}}+\|\mathcal{Q}_{2}(\mu^{h})\|^{2}_{0,\Omega_{2}}=\|{u}^{h,\mu}_{k,C}\|^{2}_{0,\Omega_{k}}\,.$
(34)

Next, we recall the equivalence relation [63, p.386, Lemma 9.7]

$\underline{C}_{\nu}h^{\nu}|\bm{v}|^{2}\leq\|{v}^{h}\|^{2}_{0,\omega}\leq\overline{C}_{\nu}h^{\nu}|\bm{v}|^{2}$
(35)

that holds for any nodal finite element function ${v}^{h}$ defined on a quasi-
uniform finite element partition of a bounded region
$\omega\subset\mathbb{R}^{\nu}$ and its coefficient vector
$\bm{v}\in\mathbb{R}^{n}$. Application of the upper bound in (35) to the
lifting ${u}^{h,\mu}_{k,C}$, together with the definition (32) of its
coefficient vector, yields

$\|{u}^{h,\mu}_{k,C}\|^{2}_{0,\Omega_{k}}\leq\overline{C}_{\nu}h^{\nu}_{k}|\bm{u}^{\mu}_{k,C}|^{2}=\overline{C}_{\nu}h^{\nu}_{k}|\bm{\mu}|^{2}\,.$

Since $\bm{\mu}$ is also the coefficient of the Lagrange multiplier $\mu^{h}$,
application of the lower bound in (35) with $\nu-1$ gives the inequality

$|\bm{\mu}|^{2}\leq\frac{1}{\underline{C}_{\nu-1}h^{(\nu-1)}_{k}}\|\mu^{h}\|^{2}_{W}\,.$

In conjunction with (34), these two inequalities combine to produce the bound

$\|\mathcal{Q}(\mu^{h})\|^{2}_{V}\leq\frac{\overline{C}_{\nu}}{\underline{C}_{\nu-1}}h_{k}\|\mu^{h}\|^{2}_{W}.$

Therefore, the second assertion of the lemma holds with
$C_{2}=\overline{C}_{\nu}/\underline{C}_{\nu-1}$ and $\alpha=1/2$. ∎

Having established the existence of the operator $\mathcal{Q}$, the proof of
the inf-sup condition is straightforward. The following lemma provides the
formal argument.

######  Lemma 2.

Assume that $h_{k}<1$. Then, the bilinear form $b(\cdot,\cdot)$ satisfies the
inf-sup condition (30).

###### Proof.

Let $\mu^{h}\in\widetilde{W}^{h}$ be an arbitrary reduced space function.
Using the assumption on the mesh size and the properties (31) of the operator
$\mathcal{Q}$, we find that

$\sup_{\\{{v}^{h}_{1,C},{v}^{h}_{2,C}\\}\in\widetilde{V}^{h}_{C}\times\widetilde{V}^{h}_{C}}\frac{b({v}^{h}_{1,C},{v}^{h}_{2,C};{\mu}^{h})}{\|\\{{v}^{h}_{1,C},{v}^{h}_{2,C}\\}\|_{V}}\geq\frac{b(\mathcal{Q}(\mu^{h}),\mu^{h})}{\|\mathcal{Q}(\mu^{h})\|_{V}}\overset{(\ref{eq:Q}a)}{=}\frac{\|\mu^{h}\|^{2}_{W}}{\|\mathcal{Q}(\mu^{h})\|_{V}}\overset{(\ref{eq:Q}b)}{\geq}\frac{\underline{C}_{\nu-1}}{\overline{C}_{\nu}}h^{-1/2}_{k}\|\mu^{h}\|_{W}\geq\beta\|\mu^{h}\|_{W},$

with $\beta=\underline{C}_{\nu-1}/\overline{C}_{\nu}$. ∎

######  Remark 8.

The proofs of Lemma 1 and Lemma 2 highlight the key role played by the _trace-
compatibility_ condition for our analysis. Specifically, this condition
guarantees the existence of a lifting (32) of the Lagrange multiplier into the
composite RB space, which is needed for the construction of the operator
$\mathcal{Q}$. This operator is essential for showing that (30) holds.

This completes the verification of the assumptions necessary to assert that
the Schur complement (24) is SPD. Therefore, we can conclude that the IVR
formulation for the composite coupled ROM-ROM problem is well-posed.

###  7.2 Composite reduced basis coupled ROM-FOM

Let us now specialize the results of Section 7.2 to the case of the coupled
ROM-FOM formulation (21) (note that the FOM-ROM case, where a FOM is used in
$\Omega_{1}$ and a ROM is used in $\Omega_{2}$, is analogous). To prove that
the Schur complement

$\widehat{S}:=\widehat{G}_{1}\widetilde{M}_{1}^{-1}\widehat{G}_{1}^{T}+\widehat{G}_{2}{M}_{2,D}^{-1}\widehat{G}_{2}^{T}$
(36)

is symmetric and positive definite, we will use the same variational approach
based on showing that the auxiliary mixed variational form (25) satisfies the
conditions of Brezzi’s theory. To that end, we specialize the functional
setting from Section 7.1 to the present case as follows.

First, we shall retain the norms (26) for the full order state space
$V^{h}_{D}$ and the Lagrange multiplier space $W^{h}$. Next, we define the
_hybrid_ state space $\widetilde{V}^{h}_{H}\subset V^{h}_{D}$ as

$\widetilde{V}^{h}_{H}=\widetilde{S}^{h}_{1,C}\times{S}^{h}_{2,D},$

where
$\widetilde{S}^{h}_{1,C}=\widetilde{S}^{h}_{1,\gamma}\oplus\widetilde{S}^{h}_{1,0}$
is the composite RB space on $\Omega_{1}$. Finally, we set

$\widehat{W}^{h}=\left\\{\begin{array}[]{ll}\widetilde{G}^{h}_{1,\gamma},&\mbox{for
option rLM}\\\\[4.30554pt] G^{h}_{2},&\mbox{for option
fLM}\end{array}\,.\right.$

As in the case of (17), it is straightforward to show that restriction of the
auxiliary form (25) to $\widetilde{V}^{h}_{H}$ and $\widehat{W}^{h}$ produces
the matrix on the left-hand side of (21). Likewise, since
$\widetilde{V}^{h}_{H}$ is a subspace of $V^{h}_{D}$, the first Brezzi
condition is trivially satisfied.

Specialized to (21), the second (inf-sup) condition now reads: for any
$\widehat{\mu}^{h}\in\widehat{W}^{h}$ the form $b(\cdot,\cdot)$ satisfies the
inequality

$\sup_{\\{{v}^{h}_{1,H},{v}^{h}_{2,H}\\}\in\widetilde{V}^{h}_{H}\times\widetilde{V}^{h}_{H}}\frac{b({v}^{h}_{1,H},{v}^{h}_{2,H};{\mu}^{h})}{\|\\{{v}^{h}_{1,H},{v}^{h}_{2,H}\\}\|_{V}}\geq\beta\|\widehat{\mu}^{h}\|_{W},$
(37)

with a mesh-independent constant $\beta$. Again, as in the case of (17), the
proof of (37) requires an operator
$\mathcal{Q}:\widehat{W}^{h}\mapsto\widetilde{V}^{h}_{H}$ such that for every
$\widehat{\mu}^{h}\in\widehat{W}^{h}$ there holds

$(a)\ \ \|\widehat{\mu}^{h}\|^{2}_{W}\leq
C_{1}b(\mathcal{Q}(\widehat{\mu}^{h});\widehat{\mu}^{h})\quad\mbox{and}\quad(b)\
\ \|\mathcal{Q}(\widehat{\mu}^{h})\|_{V}\leq
C_{2}\widehat{h}^{\alpha}_{\gamma}\|\widehat{\mu}^{h}\|_{W}\,,$ (38)

where $\widehat{h}_{\gamma}=h_{1,\gamma}$ for option rLM,
$\widehat{h}_{\gamma}=h_{2,\gamma}$ for option fLM, $\alpha\geq 0$, and
$C_{1}$, $C_{2}$ are positive constants independent of $\widehat{h}_{\gamma}$.

Since both options for the Lagrange multiplier space are trace-compatible, the
operator $\mathcal{Q}$ for the coupled ROM-FOM case can be easily defined by a
minor modification of (33) to account for the particular option used.
Specifically, we set

$\mathcal{Q}(\mu^{h})=\left\\{\begin{array}[]{rl}\\{{u}^{h,\mu}_{1,C},0\\}\in\widetilde{V}^{h}_{C}&\mbox{for
option rLM}\\\\[6.45831pt]
-\\{0,{u}^{h,\mu}_{2,D}\\}\in\widetilde{V}^{h}_{C}&\mbox{for option
fLM}\end{array}\right.\,.$ (39)

where ${u}^{h,\mu}_{1,C}$ is the lifting of $\widehat{\mu}^{h}$ defined in
Lemma 1 and ${u}^{h,\mu}_{2,D}$ is the lifting of $\widehat{\mu}^{h}$ into the
finite element space $S^{h}_{2,D}$ defined by the coefficient vector
$\bm{u}^{\mu}_{2,D}=(\widehat{\bm{\mu}},\bm{0})$. It is straightforward to
verify that the operator (39) satisfies the inequalities in (38). Then, using
the same arguments as in the proof of Lemma 2, one can show that (37) holds
with the same constant $\beta$ as in that lemma. This establishes all
conditions necessary for the Schur complement (36) to be SPD.

A few comments about these results are now in order. As we have mentioned
earlier, it is possible to prove the full column rank property of the
transpose constraint matrices in the coupled ROM-ROM and ROM-FOM problems
directly using purely algebraic tools. However, this approach fails to account
for the fact that we are dealing with matrices obtained through a
discretization process followed by a Galerkin projection. Such matrices carry
an implicit dependence on the discretization mesh parameter and the size of
the RBs employed in the projection. As a result, the condition numbers and the
ranks of the matrices in the coupled FOM-FOM, ROM-ROM, and ROM-FOM problems
also depend on these parameters. An algebraic approach treats these matrices
as having a given fixed dimension and generally cannot reveal the dependence
of condition numbers and ranks on the mesh size and the RB dimension.

In contrast, the inf-sup condition not only establishes that these transpose
constraint matrices have full column ranks, but it also provides a lower bound
on their smallest singular values; see, e.g., [64]. In particular, by showing
that the inf-sup conditions (30) and (37) hold with mesh and RB-independent
lower bounds, we effectively prove that the smallest singular values of the
associated constraint matrices are bounded away from zero _independently_ of
the mesh size and/or the dimensions of the reduced bases. To put it
differently, by adopting a variational approach we are able to show that the
transpose constraint matrices cannot become computationally rank-deficient
both when one varies the mesh size of the coupled FOM-FOM and when one varies
the dimension of the composite RB. This property is highly non-trivial to
establish using algebraic approaches.

##  8 Numerical results

The objectives of this section are two-fold. First, we aim to confirm
numerically the theoretical analysis in Section 7, specifically the fact that
projection of the coupled FOM-FOM (4) onto the composite RB spaces, using a
trace-compatible Lagrange multiplier space, leads to coupled ROM-ROM (17) and
ROM-FOM (21) problems with non-singular Schur complements, independently of
the RB size. Our second goal is to demonstrate numerically the accuracy of the
partitioned schemes in the two distinct simulation settings outlined in
Section 1. We recall that the first one is characterized by a continuous
diffusion coefficient, i.e., we consider (1) with $\kappa_{1}=\kappa_{2}$.
Keeping in mind the distinctions with the use of this term in the ROM
literature highlighted in Section 1.2, we shall refer to this case as the
“domain decomposition” (DD) setting. We also recall that the second setting
represents a bona fide transmission problem (TP) characterized by a
discontinuous diffusion coefficient. Section 8.1 presents reproductive and
predictive results for the DD setting, while Section 8.2 provides predictive
tests in the TP setting. Since reproductive tests for the latter largely
mirror the results for former, Section 8.2 includes only predictive TP tests.

Following our previous work [4], we use the solid body rotation test from
[65], specialized to (1). The computational domain for this test is the unit
square $\Omega:=(0,1)\times(0,1)$, the initial condition comprises a cone, a
cylinder, and a smooth hump (Figure 4(a)), and the advection field is defined
as $\bm{a}:=(0.5-y,x-0.5)$. We split $\Omega$ into subdomains
$\Omega_{1}:=(0,0.5)\times(0,1)$ and $\Omega_{2}:=(0.5,1)\times(0,1)$, impose
homogeneous Dirichlet boundaries on all non-interface boundaries $\Gamma_{i}$,
for $i=1,2$, and set the final time to be $T_{f}:=2\pi$, representing one full
rotation.

In all examples we use a uniform partition of $\Omega$ into $64\times 64$
square elements yielding 4225 nodes in $\Omega$ and 2145 nodes in $\Omega_{i}$
for $i=1,2$, as seen in Figure 4(b). It is easy to see that
$\gamma^{h}_{1}=\gamma^{h}_{2}$, i.e., the interface finite element partitions
induced by the subdomain meshes are identical. This setting eliminates error
pollution due to non-matching interface grids from the numerical results and
allows us to examine the “pure” properties of the partitioned schemes. In
particular, in this setting, the IVR solution of the coupled FOM-FOM problem
(4) obtained by solving the subdomain equations in (7) coincides, to machine
precision, with a _single domain solution_ obtained by treating (1) as a
single PDE with a discontinuous coefficient; see [1]. Finally, we note that
all results in this section were obtained by using the forward Euler method as
the time discretization scheme.

(a) Initial conditions

(b) The finite element partitions $\Omega^{h}_{1}$ (blue) and $\Omega^{h}_{2}$
(red).

Figure 4: Initial conditions, domain partitioning, and mesh for the model 2D
transmission problem.

###  8.1 Domain decomposition setting

The model problem is parameterized with respect to the diffusion coefficient
$\kappa_{i}$, which for the domain decomposition case is the same on both
subdomains, i.e., $\kappa_{1}=\kappa_{2}:=\kappa$. We perform the reproductive
tests using a RB obtained from solution snapshots corresponding to
$\kappa=10^{-5}$. For the predictive tests, we define the reduced bases from
snapshots computed with $\kappa=10^{-2}$ and $\kappa=10^{-8}$, and then
simulate the model problem with $\kappa=10^{-5}$.

To obtain the subdomain solution snapshots, we restrict a single domain finite
element solution of (1) to $\Omega_{1}$ and $\Omega_{2}$, respectively. The
single domain solution is computed using the time step $\Delta
t_{s}=9.156\times 10^{-4}$ for $\kappa=10^{-2}$, and $\Delta t_{s}=1.684\times
10^{-3}$ for $\kappa=10^{-5}$ and $10^{-8}$. These time steps are determined
from the Courant-Friedrichs-Lewy (CFL) condition. Since both the reproductive
and the predictive tests are performed for $\kappa=10^{-5}$, the partitioned
solution scheme employed for all coupled formulations uses the time step
$\Delta t=1.684\times 10^{-3}$. To demonstrate the importance of the composite
RB and trace-compatible Lagrange multipliers for the properties of the Schur
complement, we present results for the partitioned solution of the coupled
ROM-ROM and FOM-ROM problems, implemented with the composite RB and with
alternative choices for the Lagrange multiplier (LM) space. We use as a
benchmark the single domain solution introduced earlier. For the coupling of a
ROM to a FOM we choose to implement the FOM on $\Omega_{1}$ and the ROM on
$\Omega_{2}$, but note that similar performance is achieved if this choice
were reversed. Still, in what follows, for consistency, we label this
formulation as FOM-ROM. To summarize, we perform tests using the following
schemes:

  * 1.

RR-rLM: partitioned solution of the coupled ROM-ROM problem (17).

  * 2.

FR-fLM: partitioned solution of the coupled FOM-ROM (21) with the (full) LM
space $G^{h}_{1}$.

  * 3.

FR-rLM: partitioned solution of the coupled FOM-ROM (21) with the (reduced) LM
space ${\color[rgb]{0,0,0}{\Phi}}_{2,\gamma}$.

  * 4.

FF-fLM partitioned solution of the coupled FOM-FOM (4).

The partitioned schemes above are supported by rigorous theory that asserts
the existence of well-posed Schur complements for the associated coupled
problems, i.e., Schur complements that are provably non-singular and have
bounded condition numbers. As an example of a formulation that is not
supported by such a theory, we consider

  * 1.

RR-fLM: partitioned solution of the coupled ROM-ROM problem (17) with the
(full) LM space $G^{h}_{1}$.

####  8.1.1 Reproductive results

First, we present the results for the reproductive case. With the snapshot
time step set to $\Delta t_{s}=1.684\times 10^{-3}$, 3732 snapshots are
collected. A prerequisite for an effective ROM is the rapid decay of the
singular values. We first confirm that this is indeed the case and that most
of the energy, defined in (9), is contained within a much smaller subset of
the snapshots.

Figure 5: Snapshot energy (9) as a function of interior
${\color[rgb]{0,0,0}{\Phi}}_{i,0}$ and interface
${\color[rgb]{0,0,0}{\Phi}}_{i,\gamma}$ basis sizes in the reproductive
regime.

The plots in Figure 5 show the energy retained in the interior
${\color[rgb]{0,0,0}{\Phi}}_{i,0}$ and interface
${\color[rgb]{0,0,0}{\Phi}}_{i,\gamma}$ RB sets as a function of their
respective sizes, $d_{i,0}$ and $d_{i,\gamma}$. The plot reveals that just
$d_{1,0}=24$, $d_{2,0}=21$ and $d_{i,\gamma}=6$ interior and interface modes,
respectively, are sufficient to capture 99% of the energy in $X_{i,0}$ and
$X_{i,\gamma}$. Setting $d_{1,0}=57$, $d_{2,0}=50$, and $d_{i,\gamma}=18$
captures 99.999% of the snapshot energies. In what follows, we denote the size
of the composite RB
${\color[rgb]{0,0,0}{\Phi}}_{i,C}=\\{{\color[rgb]{0,0,0}{\Phi}}_{i,\gamma},{\color[rgb]{0,0,0}{\Phi}}_{i,0}\\}_{i=1,2}$
as $d_{i,C}=d_{i,0}+d_{i,\gamma}$.

To assess the accuracy of the partitioned solutions of the coupled ROM-ROM and
FOM-ROM problems, we report their relative errors with respect to the single
domain solution of the model problem with the same diffusion coefficient as
used for the reproductive tests, i.e., $\kappa=10^{-5}$. We define these
errors as

$\displaystyle{\color[rgb]{0,0,0}{\epsilon(t):=\frac{||\\{\bm{u}^{t}_{1,P},\bm{u}^{t}_{2,P}\\}-\\{\bm{u}^{t}_{1,S},\bm{u}^{t}_{2,S}\\}||_{V}}{||\\{\bm{u}^{t}_{1,S},\bm{u}^{t}_{2,S}\\}||_{V}},}}$
(40)

where $\|\cdot\|_{V}$ is the norm defined in (26),
${\color[rgb]{0,0,0}{\\{\bm{u}^{t}_{1,S},\bm{u}^{t}_{2,S}\\}}}$ is the single
domain solution of the model problem, and
${\color[rgb]{0,0,0}{\\{\bm{u}^{t}_{1,P},\bm{u}^{t}_{2,P}\\}}}$ denotes a
partitioned solution of the coupled ROM-ROM, FOM-ROM or FOM-FOM problems, at a
chosen time $t\in[0,2\pi]$.

When using the composite RB, one can set the dimensions $d_{i,0}$ and
$d_{i,\gamma}$ for the interior and interface bases independently. Here, we
choose $d_{i,\gamma}$ to be two-fifths of the total composite basis size
$d_{i,C}$, i.e., we set

$d_{i,\gamma}=\frac{2}{5}\Big{(}d_{i,\gamma}+d_{i,0}\Big{)}\implies
d_{i,\gamma}=\frac{2}{3}d_{i,0}.$ (41)

Note that the dimension of the reduced Lagrange multiplier space in the
coupled ROM-ROM (17) is given by either $d_{1,\gamma}$ or $d_{2,\gamma}$.

When setting $d_{i,\gamma}$, one also has to account for the fact that the
total number of modes $d^{\max}_{i,\gamma}$ available for the construction of
the interface RB ${\color[rgb]{0,0,0}{\Phi}}_{i,\gamma}$ is, in general,
smaller than the number $d^{\max}_{i,0}$ available for the construction of
${\color[rgb]{0,0,0}{\Phi}}_{i,0}$. As a result, direct application of (41)
may yield values for $d_{i,\gamma}$ that exceed the number
$d^{\max}_{i,\gamma}$ of interface modes available. To avoid this, we further
refine the choice of the interface dimension according to

$d_{i,\gamma}=\min\Big{\\{}\frac{2}{3}d_{i,0},d^{\max}_{i,\gamma}\Big{\\}}.$

In all our simulations $d^{\max}_{i,\gamma}=63$.

Figure 6: Relative error (40) at the final time $T_{f}=2\pi$ of the
partitioned solution for each coupled formulation as a function of the
composite reduced basis size $d_{i,C}=d_{i,0}+d_{i,\gamma}$ in the
reproductive regime.

(a) $d_{i,0}=15,d_{i,\gamma}=10$

(b) $d_{i,0}=60,d_{i,\gamma}=40$

(c) $d_{i,0}=90,d_{i,\gamma}=60$

(d) $d_{i,0}=237,d_{i,\gamma}=63$

(e) $d_{i,0}=1953,d_{i,\gamma}=63$

(f) Single domain ROM

Figure 7: Relative error (40) of the partitioned solution for select RB sizes
as a function of time in the reproductive regime.

We first examine the behavior of the relative error (40) when our partitioned
schemes are applied to coupled formulations with provably well-posed Schur
complements. Figure 6 plots $\epsilon(T_{f})$, i.e., the relative error at the
final time, as a function of the composite RB size $d_{i,C}$. The plots in
this figure show that as the total size of the composite RB $d_{i,C}$ is
increased, the error at the final time in the partitioned solutions of the
coupled ROM-ROM and FOM-ROM problems approaches that of the partitioned
solution of the coupled FOM-FOM problem, as expected for a reproductive test.
At the same time, we observe that, while $\epsilon(T_{f})$ for the coupled
FOM-ROM problem with the reduced space Lagrange multiplier is essentially the
same as for the other formulations when $d_{i,C}$ is large enough, it is
significantly higher for small RB sizes.

To examine this issue further we plot the relative error _as a function of
time_ for “small” ($d_{i,C}\leq 100$, Fig. 7(a,b)), “medium”
($15{\color[rgb]{0,0,0}{0}}\leq d_{i,C}\leq 300$, Fig. 7(c,d)), and “large”
($d_{i,C}=2016$, Fig. 7(e)) composite RB sizes. Note that the “large” RB has
the same number of modes as the FOM. For comparison, in Fig. 7(f) we plot
$\epsilon(t)$ for three instances of a single domain ROM. The three instances
have approximately the same total number of modes, i.e.,
$d_{1,0}+d_{2,0}+d_{i,\gamma}$, as the “small”, “medium” and “large” RBs in
Fig. 7(b), (d), and (e), respectively.

The plots in Figure 7 clearly show that for “medium” and “large” RB sizes
$\epsilon(t)$ for all coupled formulations is stable and roughly comparable to
that of the single domain ROM. However, the behavior of $\epsilon(t)$ for the
FR-rLM formulation deviates significantly from that of the other formulations
when the RB size is small. The plots in Fig. 7(a,b) show that, up to $t\approx
2$, the relative errors of all formulations have roughly the same magnitude.
However, as time integration continues past this time, $\epsilon(t)$ for the
FR-rLM formulation begins to grow, while the relative error of all other
formulations remains about the same. Therefore, the large size of
$\epsilon(T_{f})$ for this formulation is caused by the accumulation of errors
during the explicit time stepping.

Although a rigorous analysis of the source of these errors is beyond the scope
of this paper, below we offer some insights into the probable cause for the
growth of $\epsilon(t)$ for the FR-rLM formulation with a “small” RB size.
Before we provide the details, let us remark that the behavior of
$\epsilon(t)$ in this case does not contradict the analysis in Section 7
because our theory asserts well-posedness of the Schur complement (24), which
is _independent_ of time. In fact, the plots in Figure 8, that will be
discussed in more detail shortly, reveal that for “small” RB sizes condition
number of the Schur complement for the FR-rLM formulation is actually lower
than that for the benchmark coupled FOM-FOM problem.

Since FR-rLM and FR-fLM only differ in the choice of the Lagrange multiplier
space, let us compare and contrast the enforcement of the coupling condition
(2) in these formulations. This task is greatly simplified by the fact that in
all our examples $\gamma_{1}^{h}={\color[rgb]{0,0,0}{\gamma_{2}^{h}}}$. As a
result, $S^{h}_{1,\gamma}=S^{h}_{2,\gamma}$,
$n_{1,\gamma}=n_{2,\gamma}=n_{\gamma}$, and
$G_{1,\gamma}=G_{2,\gamma}=G_{\gamma}$, where $G_{\gamma}$ is a symmetric and
positive definite interface mass matrix. In the FR-fLM formulation
$\lambda^{h}\in G^{h}_{1}$. Taking into account that the FOM is defined on
$\Omega_{1}$ and that $G_{1,\gamma}=G_{2,\gamma}=G_{\gamma}$, the last
equation in (21) specializes to

$G_{1,\gamma}\dot{\bm{u}}_{1,\gamma}-G_{2,\gamma}\Phi_{2,\gamma}\dot{\widetilde{\bm{u}}}_{2,\gamma}=G_{\gamma}\left(\dot{\bm{u}}_{1,\gamma}-\Phi_{2,\gamma}\dot{\widetilde{\bm{u}}}_{2,\gamma}\right)=0\,.$
(42)

Since $G_{\gamma}$ is non-singular, it follows that

$\dot{\bm{u}}_{1,\gamma}-\Phi_{2,\gamma}\dot{\widetilde{\bm{u}}}_{2,\gamma}=0\,.$
(43)

Thus, in the FR-fLM formulation on _matching interface grids_ the coupling
condition is enforced _pointwise_.

In contrast, in the FR-rLM formulation
$\bm{\lambda}=\Phi_{2,\gamma}\widetilde{\bm{\lambda}}$ and the last equation
in (21) now assumes the form

$\Phi_{2,\gamma}^{T}G_{1,\gamma}\dot{\bm{u}}_{1,\gamma}-\Phi_{2,\gamma}^{T}G_{2,\gamma}\Phi_{2,\gamma}\dot{\widetilde{\bm{u}}}_{2,\gamma}=\Phi_{2,\gamma}^{T}G_{\gamma}\left(\dot{\bm{u}}_{1,\gamma}-\Phi_{2,\gamma}\dot{\widetilde{\bm{u}}}_{2,\gamma}\right)=0\,.$
(44)

Let $\Phi_{2,\gamma}^{\prime}$ be the $n_{\gamma}\times r_{2}-d_{2,\gamma}$
matrix of discarded left singular vectors from the SVD decomposition of the
snapshot set $X_{2,\gamma}$. Then, (44) permits the existence of a nonzero
vector $\bm{\delta}\in\mathbb{R}^{(r_{2}-d_{2,\gamma})}$ such that

$G_{\gamma}\left(\dot{\bm{u}}_{1,\gamma}-\Phi_{2,\gamma}\dot{\widetilde{\bm{u}}}_{2,\gamma}\right)=\Phi_{2,\gamma}^{\prime}\bm{\delta}\,.$

It follows that

$\dot{\bm{u}}_{1,\gamma}-\Phi_{2,\gamma}\dot{\widetilde{\bm{u}}}_{2,\gamma}=G_{\gamma}^{-1}\Phi_{2,\gamma}^{\prime}\bm{\delta}\,.$
(45)

In other words, in the FR-rLM formulation, the coupling condition is satisfied
approximately whereas in the FR-fLM case this condition holds pointwise.

The plots in Figure 5 reveal that just 6 interface modes are sufficient to
capture $99\%$ of the snapshot energy (9) of $X_{i,\gamma}$. However, for
small values of $d_{2,\gamma}$, the snapshot energy contained in the discarded
modes $\Phi_{2,\gamma}^{\prime}$ may still be large enough so that
accumulation of errors at each time step due to the right hand side in (45)
eventually destroys the accuracy of the numerical solution.

We note that this phenomenon is not limited to the FOM-ROM formulation, but is
rather a consequence of coupling subdomain formulations that are imbalanced
with respect to their accuracy. Indeed, we observed similar error growth when
a ROM with a “large” RB size was coupled to a ROM with a “small” RB size.
Thus, when coupling subdomain models that differ significantly in their
resolution, a useful rule of a thumb would be to define the Lagrange
multiplier space by always using the side with a higher resolution.

(a) All coupled problems

(b) Provably well-posed Schur complement

Figure 8: Condition number of Schur complement matrix for each coupled
formulation as a function of the composite reduced basis size
$d_{i,C}=d_{i,0}+d_{i,\gamma}$ size in the reproductive regime. Subfigure (a)
reports results for all methods evaluated, whereas subfigure (b) focuses only
on methods with provably well-posed Schur complements.

Next, we highlight the importance of the Lagrange multiplier basis for the
well-posedness of the Schur complement in the coupled ROM-ROM and FOM-ROM
problems. To that end, in Figure 8, we compare and contrast the condition
number of this matrix for the couplings that satisfy the trace compatibility
condition with the RR-fLM scheme that does not satisfy this condition.
Conditioning of the Schur complement is a measure of its “well-posedness” and
can be used to confirm the conclusions from the analysis in Section 7.

The most important takeaway from Figure 8 is that the Schur complements of the
coupled ROM-ROM (17) and FOM-ROM (21) problems, which employ _trace-
compatible_ Lagrange multiplier spaces conforming with the theory in Section
7, have essentially constant condition numbers666Although, in Figure 8(b), the
condition number of the Schur complement for the FR-fLM problem appears
significantly larger than that for the other couplings, the range for the
$y$-axis in Figure 8(b) is $[0,28.1]$ with the upper limit representing
$\max\rm{cond}(S)$ for the FR-fLM problem. Thus, in all cases $\rm{cond}(S)$
is of order at most $O(10)$. with respect to the reduced basis dimension. This
corroborates numerically the theoretical conclusions asserting that the
condition number of the Schur complement should be independent of the size of
the reduced basis. Moreover, we see that using the trace-compatible Lagrange
multiplier spaces required by the theory produces coupled ROM-ROM and FOM-ROM
problems whose Schur complements are of the same order as those of the coupled
FOM-FOM problem. We recall that the latter also uses trace-compatible Lagrange
multiplier spaces and is provably well-posed [1].

At the same time, using Lagrange multiplier spaces that are _not trace-
compatible_ clearly leads to Schur complements whose condition number
_depends_ on the reduced basis size. Specifically, by inspecting Figure 8(a),
we see that when the full order interface finite element space $G^{h}_{2}$ is
used as a Lagrange multiplier space to couple two ROMs (RR-fLM), the Schur
complement of the resulting coupled problem has very high condition numbers
for smaller dimensions of the reduced basis. While the condition number does
decrease as the reduced basis size increases, it is still high compared to
that of the coupled FOM-FOM problem, and it only reaches a reasonable scale
when the reduced basis size is larger than one would wish to consider.

(a) RR-rLM, 15/10 modes

(b) FR-fLM, 15/10 modes

(c) FR-rLM, 15/10 modes

(d) RR-rLM 60/40 modes

(e) FR-fLM 60/40 modes

(f) FR-rLM 60/40 modes

(g) RR-rLM 90/60 modes

(h) FR-fLM 90/60 modes

(i) FR-rLM 90/60 modes

Figure 9: Comparison of the interface states at $T_{f}$ for the partitioned
schemes with provably well-posed Schur complements vs. the single domain
(monolithic) solution of the model problem. Reproductive test in the DD
setting. The oscillations in the FR-rLM formulations with “small” RB sizes are
due to accumulation of interface errors during the time integration caused by
the approximate enforcement (45) of the coupling condition. The legend “$m/n$
modes” corresponds to $m$ interior and $n$ interface modes.

To understand the root cause for this behavior, recall that trace-
compatibility requires every element of the Lagrange multiplier space to have
a corresponding subdomain state whose trace on the interface matches the
multiplier. This property is essential for the construction of the operator
$\mathcal{Q}$ that plays a key role in showing that the Schur complement is
non-singular; see Remark 8. At the same time, it is clear that when the
subdomain states are represented by a reduced basis, their traces will not be
able to reproduce every possible element of the full interface space
$G^{h}_{2}$, i.e., the latter is not trace-compatible. As the size of the
reduced basis for the states increases, trace compatibility is restored and
the condition number of the Schur complement is reduced. This inflection point
is clearly visible in Figure 8(a) and corresponds to the instance when the
traces of the reduced basis states contain the Lagrange multiplier space. An
algebraic explanation of this behavior is that when the subdomain states are
represented by a reduced basis, the full size Lagrange multiplier space will
over-constrain the states leading to nearly rank deficient, or rank-deficient
transpose constraint matrices.

######  Remark 9.

Numerical studies in [4] have shown that the partitioned solution of the
coupled ROM-ROM problem implemented with the full LM space $G^{h}_{2}$ has, to
machine precision, the same errors as the partitioned solution of the this
problem implemented with the reduced interface LM space
${\color[rgb]{0,0,0}{\Phi}}_{i,\gamma}$. This suggests that solution errors
alone do not tell the whole story about the quality of the Schur complement
underpinning these partitioned solutions. Using Lagrange multiplier spaces
that violate our analysis may result in seemingly reasonable errors for
specific instances of discretization and ROM parameters, but is not guaranteed
to work across all possible regimes.

Finally, in Figure 9, we examine how well the partitioned solutions of the
provably well-posed coupled problems satisfy the interface condition, which is
an important characteristic of any partitioned scheme. To that end, we compare
the partitioned solutions with the single domain solution of the model problem
on the interface $\gamma$. The plots in Figure 9 reveal that, for a large
enough composite reduced basis, partitioned solutions of the coupled ROM-ROM
and FOM-ROM problems have essentially the same accuracy on the interface as
the solution of the coupled FOM-FOM problem (FF-fLM). We note that the FOM-ROM
coupling with reduced LM space does require a larger basis size for the same
accuracy as the FOM-ROM with full LM space, but it is capable of attaining the
same level of errors. In particular, as discussed earlier, the oscillations in
the FR-rLM formulations with “small” RB sizes are due to accumulation of
interface errors during the time integration caused by the approximate
enforcement (45) of the coupling condition.

####  8.1.2 Predictive results

We recall that for the predictive tests we collect snapshots at two diffusion
coefficients, $\kappa_{i}=10^{-2}$ and $10^{-8}$, and compute the partitioned
solutions with $\kappa_{i}=10^{-5}$ for $i=1,2$. We collect a total of 10,594
solution snapshots for $\kappa_{i}=10^{-2}$ and $\kappa_{i}=10^{-8}$ using the
time steps stated at the beginning of this section. Again, we note that the
singular values decay rapidly, so that we are able to capture most of the
snapshot energy within a much smaller subset of modes, as shown in Figure 10.
These plots reveal that only $d_{1,0}=26$, $d_{2,0}=22$, and $d_{i,\gamma}=7$
interior and interface modes are sufficient to capture 99% of the energy in
$X_{i,0}$ and $X_{i,\gamma}$. Setting $d_{1,0}=67$, $d_{2,0}=58$, and
$d_{i,\gamma}=20$ captures 99.999% of the snapshot energies. This is
approximately the same rate of decay as shown in the reproductive case.

Figure 10: Snapshot energy (9) as a function of interior
${\color[rgb]{0,0,0}{\Phi}}_{i,0}$ and interface
${\color[rgb]{0,0,0}{\Phi}}_{i,\gamma}$ basis sizes in the predictive regime.

We first consider the relative errors (40) of the partitioned solutions of the
coupled ROM-ROM and FOM-ROM problems. Figures 11–12 summarizes our results. In
all three cases we see that, for a sufficiently large composite reduced basis
size, partitioned solutions are able to achieve relative errors of roughly
$10^{-2}$ or $10^{-3}$. This confirms the ability of the partitioned schemes
presented in this paper to simulate the model problem for parameter values
that have not been used in the construction of the reduced basis.

In the predictive case, as previously in the reproductive case, the
approximate satisfaction of the coupling condition (45) leads to accumulation
of errors during the time integration that eventually destroys the accuracy of
the solution for “small” RB sizes. As a result, as in the reproductive test,
the relative error at the final time for the FR-rLM formulation is
significantly larger than that for the other formulations. The plots in Figure
12 reveal that, also similar to the reproductive case, all formulations have
comparable errors up to time $t\approx 2$, and that the error buildup for the
FR-rLM formulation begins after that time.

We next examine the condition numbers of the Schur complement matrices
involved in the predictive tests. Again, we compare and contrast the
conditioning of these matrices for the couplings that satisfy the trace
compatibility condition and the RR-fLM coupling that satisfies this condition
only for large enough size of the reduced basis. These results are summarized
in Figure 13. The plots in this figure mirror the behavior of the condition
number observed in the reproductive test. Thus, one can conclude that the
theoretical results in Section 7 remain in full force in the predictive regime
as well, confirming the need for trace-compatible Lagrange multiplier spaces
when constructing the coupled ROM-ROM and FOM-ROM problems.

Figure 11: Relative error (40) at the final time $T_{f}=2\pi$ of the
partitioned solution for each coupled formulation as a function of the
composite reduced basis size $d_{i,C}=d_{i,0}+d_{i,\gamma}$ in the predictive
regime.

(a) $d_{i,0}=15,d_{i,\gamma}=10$

(b) $d_{i,0}=60,d_{i,\gamma}=40$

(c) $d_{i,0}=90,d_{i,\gamma}=60$

(d) $d_{i,0}=237,d_{i,\gamma}=63$

(e) $d_{i,0}=1953,d_{i,\gamma}=63$

(f) Single domain ROM

Figure 12: Relative error (40) of the partitioned solution for select basis
sizes as a function of time in the predictive regime. The error plots for 160
and 537 modes in subfigure (f) are indistinguishable.

(a) All coupled problems

(b) Provably well-posed Schur complement

Figure 13: Condition number of Schur complement matrix for each coupled
formulation as a function of the composite reduced basis size
$d_{i,C}=d_{i,0}+d_{i,\gamma}$ size in the predictive regime. Subfigure (a)
reports results for all methods evaluated, whereas subfigure (b) focuses only
on methods with provably well-posed Schur complements.

We conclude with results showing how well the partitioned solutions satisfy
the interface condition in the predictive tests. As in the reproductive test,
we focus only on the well-posed coupled formulations, and compare the
interface states of the partitioned and single domain solutions. These results
are summarized in Figure 14(f). We again see that partitioned solutions of the
couplings with provably well-posed Schur complements are able to satisfy the
interface condition. The coupled ROM-ROM and FOM-ROM problems display
agreement with the single domain solution on the interface. Not surprisingly,
we see that, as in the reproductive test, the FR-rLM formulation requires a
larger basis set to achieve the same accuracy as the other formulations. In
particular, for “small” RB sizes we see the same oscillatory behavior at the
final time, which is caused by the approximate enforcement (45) of the
coupling condition and the subsequent accumulation of errors during the time
integration.

(a) RR-rLM, 15/10 modes

(b) FR-fLM, 15/10 modes

(c) FR-rLM, 15/10 modes

(d) RR-rLM 60/40 modes

(e) FR-fLM 60/40 modes

(f) FR-rLM, 60/40 modes (g) RR-rLM, 90/60 modes (h) FR-fLM, 90/60 modes (i)
FR-rLM 90/60 modes Figure 14: Comparison of the interface states at $T_{f}$
for the partitioned schemes with provably well-posed Schur complements vs. the
single domain (monolithic) solution of the model problem. Predictive test in
the DD setting. The oscillations in the FR-rLM formulations with “small” RB
sizes are due to accumulation of interface errors during the time integration
caused by the approximate enforcement (45) of the coupling condition. The
legend “$m/n$ modes” corresponds to $m$ interior and $n$ interface modes.

###  8.2 Transmission problem case

For the TP variant of our model problem, we parameterize the model problem
using a discontinuous diffusion coefficient, i.e., $\kappa_{1}\neq\kappa_{2}$.
Since in our experiments we observed essentially the same behavior as in the
DD case, we limit ourselves to showing results for a predictive TP test. For
this test, we consider a diffusion coefficient $\kappa_{1}=10^{-5}$ in
$\Omega_{1}$, and $\kappa_{2}=10^{-4}$ in $\Omega_{2}$. To obtain the
subdomain snapshots, we proceed as in the DD case and restrict a single domain
finite element solution to $\Omega_{1}$ and $\Omega_{2}$, respectively. The
single domain solution is computed using the time step $\Delta
t_{s}=1.684\times 10^{-3}$.

The snapshot energy plots in Figure 15 confirm that the rapid decay of the
singular values continues to hold in the TP case. In particular, just
$d_{1,0}=23$, $d_{2,0}=19$ and $d_{i,\gamma}=5$ interior and interface nodes,
respectively are sufficient to capture $99\%$ of the energy in $X_{i,0}$ and
$X_{i,\gamma}$.

Figure 15: Snapshot energy (9) as a function of the POD basis size for the
interior ${\color[rgb]{0,0,0}{\Phi}}_{i,0}$ and interface
${\color[rgb]{0,0,0}{\Phi}}_{i,\gamma}$ bases in the predictive regime for the
multiphysics example.

Next, in Figures 16 and 17, we show the relative errors for each formulation.
Similar to the DD case, the relative error at the final time of the FR-rLM
formulation for small RB sizes is much larger than that for the other
formulations. The plots in Figure 17 confirm that, again, this is due to the
accumulation of error during the explicit time stepping. Interestingly enough,
these plots also show that the growth of the relative error in the TP case
occurs at roughly the same time instances as in the DD case. Moreover, the
reader can observe convergence with basis refinement for all proposed schemes
(Figure 17).

We continue with plots of the condition number of the Schur complement for all
coupled formulations in Figure 18. These plots mirror the behavior of this
quantity from the DD case and once again underscore the importance of using
Lagrange multiplier spaces that satisfy the trace compatibility condition; see
Remark 8. Most notably, Figure 18(b) shows no substantive difference in the
behavior of the condition number in the TP case as long as the formulation
satisfies the trace compatibility condition.

Finally, in Figure 19, we take a look at the interface states of all
formulations with provably well-posed Schur complements at the final time
$T_{f}$. The results in this figure affirm yet again the consistency in the
behavior of the proposed schemes for the DD and TP examples.

Figure 16: Relative error (40) at $T_{f}=2\pi$ of the partitioned solution for
each coupled formulation as a function of the composite reduced basis size
$d_{i,C}=d_{i,0}+d_{i,\gamma}$ in the predictive regime for the TP example.

(a) $d_{i,0}=15,d_{i,\gamma}=10$

(b) $d_{i,0}=60,d_{i,\gamma}=40$

(c) $d_{i,0}=90,d_{i,\gamma}=60$

(d) $d_{i,0}=237,d_{i,\gamma}=63$

(e) $d_{i,0}=1953,d_{i,\gamma}=63$