# A short perspective on a posteriori error control and adaptive
discretizations

Roland Becker  LMAP, University of Pau, IPRA BP 1155, Av. de l’Université,
64013 Pau, France  Stéphane P.A. Bordas  Institute for Computational
Engineering, Faculty of Science, Technology and Communication, University of
Luxembourg, Luxembourg  Franz Chouly  University of the Republic, Faculty of
Science, Center of Mathematics, 11400 Montevideo, Uruguay  Pascal Omnes
Université Sorbonne Paris Nord, LAGA, CNRS UMR 7539, Institut Galilée, 99 Av.
J.-B. Clément, 93430, Villetaneuse, France  Université Paris-Saclay, CEA,
Service de Génie Logiciel pour la Simulation, 91191, Gif-sur-Yvette, France

###### Abstract

Error control by means of a posteriori error estimators or indicators and
adaptive discretizations, such as adaptive mesh refinement, have emerged in
the late seventies. Since then, numerous theoretical developments and
improvements have been made, as well as the first attempts to introduce them
into real-life industrial applications. The present introductory chapter
provides an overview of the subject, highlights some of the achievements to
date and discusses possible perspectives.

This preprint corresponds to the Chapter 1 of volume 58 in AAMS, Advances in
Applied Mechanics (to appear).

Keywords : error control ; a posteriori error estimators ; adaptive
discretizations ; mesh refinement ; applications.

##  1 Introduction

The numerical solution of partial differential equations is a crucial aspect
of computational science and engineering, finding applications in diverse
fields of physics and engineering. However, achieving accurate and efficient
solutions to partial differential equations is often challenging and a
posteriori error estimators are crucial tools that evaluate the accuracy and
enhance computational efficiency of numerical methods. Unlike a priori error
estimators, which are not computable, these estimators assess solutions after
computation, providing a pragmatic evaluation of accuracy.

A posteriori estimators work as evaluative tools, systematically analysing
error distributions to offer insight into numerical approximation fidelity.
This shift from anticipatory to retrospective assessment enables practitioners
to iteratively refine solutions to align them more closely with the intricate
details of the underlying mathematical models.

This article explores the conceptual landscape of a posteriori error
estimators, examining their theoretical foundations, applications, and
consequential impact on partial differential equations problem-solving. The
discussion invites an exploration of numerical accuracy, navigating the
terrain where mathematical abstraction converges with computational reality.

##  2 A few basic notions

Before entering the core of the topic, let us describe a few useful basic
notions, in a very general setting and without entering into details. For
reviews about a posteriori error estimation, see, e.g., Eriksson and Johnson
(1985); Carstensen and Merdon (2010); Nochetto and Veeser (2012); Chamoin and
Legoll (2023).

###  2.1 Numerical approximation of a mathematical model

For several decades, computers have been used in routine calculations to
provide an approximate solution to the sophisticated mathematical models in
current engineering practice, which are, for example, (linear or non-linear)
partial differential equations supplemented by (linear or non-linear) boundary
conditions and by initial conditions, and possibly coupled to other equations,
and now, to data (even in real-time).

Let us call $u$ the exact solution of this mathematical model, which belongs
to a set $S$ of admissible solutions. There are now many techniques for
transforming the original equation through a sophisticated pipeline, so that,
in the end, a concrete approximation to the solution $u$ is provided by a
computer in the form of a collection of numbers
$\left(U_{1},\ldots,U_{N}\right)\in\mathbb{R}^{N}$ which can be used to
recover a function $u_{\mathrm{computer}}(U_{1},\ldots,U_{N})$ (hopefully)
close enough to $u$.

Figure 1 and Figure 2 below provide an overview of the process of mathematical
modelling.

Figure 1: Mathematical modelling and sources of error. This figure showcases
the steps involved in mathematical modelling, which is a process used to
represent real-world phenomena using mathematical equations. The diagram
highlights the different stages of mathematical modelling, including problem
formulation, model development, parameter estimation, model validation, and
prediction. Additionally, the figure identifies potential sources of error
that may impact the accuracy and reliability of the mathematical model. These
sources of error can arise from various factors, such as measurement
uncertainties, assumptions made during model development, limitations in data
availability, simplifications in model assumptions, and uncertainties in
parameter estimation. The figure serves as a visual representation of the
complexities and challenges associated with mathematical modelling,
emphasising the need for careful consideration of potential sources of errors
to ensure the robustness and validity of the model’s results. It underscores
the importance of thorough validation and verification processes to enhance
the accuracy and reliability of mathematical models, which are crucial for
decision-making, prediction, and understanding complex systems in various
fields of science, engineering, and beyond. Figure 2: This figure showcases
the process of mathematical modelling along with the identification and
estimation of potential sources of error. The diagram illustrates the steps
involved in mathematical modelling, including problem formulation, model
development, parameter estimation, model validation, and prediction.
Furthermore, the figure highlights the importance of error estimation in the
modelling process. It identifies potential sources of error, such as
measurement uncertainties, assumptions made during model development,
limitations in data availability, simplifications in model assumptions, and
uncertainties in parameter estimation. The figure emphasises the need to
account for and quantify these sources of error in order to assess the
reliability and accuracy of the mathematical model. The figure serves as a
visual representation of the comprehensive approach to mathematical modelling,
which includes not only model development but also thorough error estimation
to enhance the robustness and validity of the model’s results. It underscores
the significance of error estimation in improving the quality of mathematical
models and their applicability in various fields of science, engineering, and
beyond.

###  2.2 Galerkin methods and the discretization error

To be illustrative, we consider (Petrov-)Galerkin methods, that include a
broad class of numerical approximation techniques based on the weak form of
the mathematical model. This includes particularly the case of well-known
Finite Element Methods (Lagrange, mixed, etc), see, e.g., Brenner and Scott
(2008); Ciarlet (2002); Ern and Guermond (2021); Quarteroni and Valli (1994);
Szabó and Babuška (2021) but also many other new methods, among which:

  * •

discontinuous Galerkin (DG) methods, see for instance the pioneering works
Arnold (1982); Lesaint and Raviart (1974) and the recent monograph Di Pietro
and Ern (2012);

  * •

recent polytopal methods such as Conforming Polygonal Finite Elements, see,
e.g., Sukumar and Tabarraei (2004); Hybrid Discontinuous Galerkin (HDG), see,
e.g., Cockburn et al. (2016), Hybrid High Order (HHO) methods, see, e.g.,
Cicuttin et al. (2021); Cockburn et al. (2016); Di Pietro and Droniou (2020);
Lemaire (2021), the Weak Galerkin Method, see, e.g., Dong and Ern (2022), the
Virtual Element Method (VEM), see, e.g. Beirão da Veiga et al. (2013); Lemaire
(2021); the Smooth Finite Element Method (SFEM), see, e.g., Liu et al. (2007);
Nguyen-Xuan et al. (2008b); modified discontinuous Galerkin with static
condensation, see, e.g., Lozinski (2019);

  * •

Discontinuous Petrov Galerkin (DPG) methods, see, e.g., Demkowicz and
Gopalakrishnan (2010); Demkowicz et al. (2012);

  * •

IsoGeometric Analysis (IGA) and variants, see, e.g., Atroshchenko et al.
(2018); Cottrell et al. (2009); Nguyen et al. (2015), motivated by the link
between computer aided design and numerical simulation;

  * •

unfitted finite elements or geometrically nonconforming finite elements, where
the mesh boundary and the domain boundary do not match, as it occurs in
fictitious domain methods, the eXtended Finite Element Method (XFEM) or the
cut Finite Element Method (cutFEM): see, e.g., Bordas and Menk (2023); Burman
et al. (2015); Duprez and Lozinski (2020); Glowinski et al. (1994); Haslinger
and Renard (2009); Moës et al. (2006); Nguyen et al. (2008); Peskin (2002);

  * •

reduced basis techniques, such as Reduced Order modelling (ROM) or Proper
Orthogonal Decomposition (POD), see, e.g., Grepl et al. (2007); Hesthaven et
al. (2016); Kerfriden et al. (2011);

  * •

spectral methods, see, e.g., Bernardi and Maday (1997) or Canuto et al.
(2006);

  * •

wavelet-based discretization, see, e.g., Bertoluzza et al. (1994); Bertoluzza
(1995); Cohen and Masson (2000); Cohen et al. (2001); Monasse and Perrier
(1998).

Of course, the above list is far from exhaustive and the discussion can be
extended, with appropriate modifications, to other classes of approximation
techniques such as finite differences, finite volumes, collocation methods,
etc, which are not based on a variational paradigm.

So, for a Petrov-Galerkin method, let us suppose the exact solution $u$
satisfies (for instance) the weak problem

$u\in V:\ a(u;v)=L(v)\qquad\forall v\in W,$ (1)

where $V,W$ are some function spaces and $L$ is a linear form. The form

$a:V\times W\rightarrow\mathbb{R}$

is possibly nonlinear in the first variable.

######  Remark 1.

The above formalism (1) does not encompass nonlinear nonsmooth problems
related to variational inequalities for instance, such as contact, friction,
plasticity. However this is not critical for the present discussion, and to
see how extensions for variational inequalities can be carried out, see, e.g.,
Han (2005) or some contributions in these volumes, among which Bartels and
Kaltenbach (2024); Gustafsson (2024); Repin (2024).

.

A discrete Petrov-Galerkin method consists in approximating Problem (1) by a
simpler problem in finite dimensional vector spaces:

$u_{N}\in V_{N}:\ a_{G}(u_{N};v_{N})=L_{G}(v_{N})\qquad\forall v_{N}\in
W_{N},$ (2)

where $V_{N}$ and $W_{N}$ are _finite dimensional spaces_ of dimension $N$,
and $a_{G}$, resp. $L_{G}$, is a form that mimics $a$, resp. $L$ (in the
simplest situations, we can take $a_{G}=a$, resp. $L_{G}=L$).

######  Remark 2.

For many methods, the trial space $V_{N}$ and the test space $W_{N}$ are the
same ($V_{N}=W_{N}$) and this corresponds to standard Galerkin (or standard
Ritz-Galerkin) methods. The case where the spaces differ is often associated
with the terminology of Petrov-Galerkin methods, or non-standard
(Ritz-)Galerkin methods, see Ern and Guermond (2004). For a historical
perspective about this class of methods, see Gander and Wanner (2012) and
references therein, which emphasise W. Ritz’s outstanding contributions
related to the approximation of the spectral biharmonic problem for Chladni
figures.

As a result, $u_{N}$ can be represented in a basis of $N$ functions, and a
computer can provide the $N$ values of the components when Problem (2) is
solved.

The challenge now is to design the spaces $V_{N}$ and $W_{N}$ carefully enough
to calculate a _discrete solution_ $u_{N}$ that represents the exact solution
$u$ as accurately as possible. To measure the difference between $u$ and
$u_{N}$, the concept of discretisation error $\epsilon_{N}$ is introduced. A
natural and simple definition can be

$\epsilon_{N}(u,u_{N}):=\|u-u_{N}\|_{V},$ (3)

where $\|\cdot\|_{V}$ is the natural norm associated with the Banach or
Hilbert structure of $V$ (Sobolev norm or energy norm). Of course, other
possibilities exist motivated by practical applications, see paragraph 2.5
below.

Since $N$ is linked to the available computing resources, a practitioner may
wish to achieve the lowest possible value for the discretisation error
$\epsilon_{N}$, while keeping $N$ as small as possible. For example, and
ideally, one would like $\epsilon_{N}$ to be of the order of machine precision
($\epsilon_{N}\simeq 10^{-16}$ for double precision arithmetic), and $N$ to be
such that the solution $u_{N}$ is obtained in real time (a few milliseconds
for example). Today, even with the enormous progress made in computing power,
this objective remains a challenge, particularly for industrial applications
where the geometry and mathematical model can be very complex. In general, a
compromise has to be found between acceptable accuracy and acceptable use of
computing resources, and this compromise is highly dependent on the context
and the targeted applications. To achieve acceptable accuracy, the main
problem one encounters is that $\epsilon_{N}$ is unknown, because the exact
solution $u$ itself is unknown.

Another issue is that the solution $u_{N}$ to Problem (2) is not exactly the
solution $u_{\textrm{computer}}(U_{1},\ldots,U_{N})$ delivered by a computer.
Indeed, the code written to provide
$u_{\textrm{computer}}(U_{1},\ldots,U_{N})$ also contains other
approximations, that are mostly due to: numerical integration, iterative
solvers for nonlinear and/or linear systems, and finite precision arithmetic,
see, e.g., Ciarlet (2002); Ern and Guermond (2004) in the context of finite
element methods. This fact introduces another layer of errors, _numerical
errors_ , that can be quantified as, for instance:

$\|u_{\textrm{computer}}(U_{1},\ldots,U_{N})-u_{N}\|_{D}$

where $\|\cdot\|_{D}$ is a convenient norm to measure this in finite
dimensions (it does not have to be necessarily the same norm as for the
discretization error).

In general, it is assumed that the numerical errors are of very small
magnitude in comparison to discretization errors. However, in some situations
they need to be taken into account: for instance a gradient conjugate solver
can be stopped after a few iterations to save computation time. See for
instance Ern and Vohralík (2013) for a technique that can resolve this issue.

Finally we can get back to Figure 1 and Figure 2 in which the discretization
error and numerical error are depicted, and complemented with the model error
that encompasses all the discrepancies between the actual physical system that
needs to be modelled and the idealised mathematical model (this important
topic is far beyond the scope of this short overview).

###  2.3 Error estimators and error control

Since the late 1970s, pioneering work within the finite element community has
shown how it is possible to compute a quantity $\eta_{N}(=\eta_{N}(u_{N}))$
that depends only of the discrete solution $u_{N}$ and that is equivalent to
the discretization error:

$\eta_{N}(u_{N})\simeq\epsilon_{N}(u,u_{N}).$ (4)

Classical references are Babuška and Rheinboldt (1978), Ladeveze and Leguillon
(1983), Eriksson and Johnson (1985), Bank and Weiser (1985), and the books
Ainsworth and Oden (2000) and Verfürth (2013). For simple mathematical models,
we can take

$\eta_{N}(u_{N})=\|L_{G}(\cdot)-a_{G}(u_{N};\cdot)\|_{\star}$ (5)

where $L_{G}(\cdot)-a_{G}(u_{N};\cdot)$ is the residual associated with
Problem (2) and $\|\cdot\|_{\star}$ is a dual norm. A problem with (5) comes
from the norm $\|\cdot\|_{\star}$, that is not computable. A highly desirable
property of an estimator is not only its computability from $u_{N}$, but it
should allow for an implementation, such that the necessary time for
computation is negligible in comparison to the time needed to solve Problem
(2).

An important mathematical property to have a trustworthy estimator is its
_reliability_ : if it can not provide the exact discretization error, it
should at least provides an _upper bound_ of the discretization error, in the
sense that, there exists $C>0$ independent of $N$ and $u_{N}$, such that

$\epsilon_{N}(u,u_{N})\leq C\,\eta_{N},$ (6)

for all $u\in V$ and $u_{N}\in V_{N}$. See for instance Verfürth (1999);
Veeser and Verfürth (2009, 2012) for a discussion and details in the case of
the residual error estimate. If the constant $C$ in (6) is equal to $1$, we
have an _exact guaranteed upper bound_ :

$\epsilon_{N}(u,u_{N})\leq\eta_{N},$ (7)

which is of high practical interest since it ensures the discrete solution
$u_{N}$ is approximated with a degree of accuracy that is known. See for
instance Neittaanmäki and Repin (2004); Vohralík (2007); Braess and Schöberl
(2008) and a discussion in Verfürth (2009) about these “constant-free”
estimates. This paves the way for _certified numerical methods_ in which one
can ensure that the discretization error is below a known threshold. Indeed,
note that if (6) is satisfied only, the estimator $\eta_{N}$ can underestimate
the error for instance.

Last but not least, it is usual to define the effectivity index of an
estimator as

$\mathrm{eff}:=\frac{\eta_{N}(u_{N})}{\epsilon_{N}(u,u_{N})}.$ (8)

The above definition is motivated by (4) and is meant to describe the accuracy
of an estimator. Hopefully, it should not vary too much with $N$ and $u$, and
should be close to 1. First, if (6) holds, this implies that

$\mathrm{eff}\geq\frac{1}{C}.$ (9)

Clearly, the boundedness of the effectivity index is equivalent to the lower
bound

$\eta_{N}\leq c\,\epsilon_{N}(u,u_{N})$ (10)

for some $c>0$ independent of $u$ and $N$; we then have $\textrm{eff}\leq c$.
Lower bounds are often called effectivity and have been an important research
topic, see Verfürth (2013). In the context of the finite element method, they
generally even have a local form, see (17) .

Asymptotic exactness of an estimator refers to

$\lim_{N\to\infty}\mathrm{eff}_{N}=1.$ (11)

This property depends on $u$, the method to compute $u_{N}$ and the sequence
of spaces $V_{N}$. In special cases, estimators such as the Bank and Weiser
type, are known to yield (11), see Babuška et al. (1992); Bank and Weiser
(1985); Durán et al. (1991) for details.

###  2.4 Error estimators and mesh refinement

In addition to assess the accuracy of a simulation, error estimators are used
to iteratively improve the numerical approximation. The typical form of the
loop is

Solve $\quad\rightarrow\quad$ Estimate $\quad\rightarrow\quad$ Mark
$\quad\rightarrow\quad$ Adapt $\quad\rightarrow\quad\cdots$

where Solve refers to solution of the discrete problem for given approximation
spaces, see, e.g., Problem (2), and the estimator is then used to adapt the
approximation spaces. The algorithm produces a sequence of meshes
$\mathcal{K}_{\ell}$ and associated spaces $V_{\ell}$, solutions $u_{\ell}$
and estimators $\eta_{\ell}$. The global upper bound justifies the use of the
estimator $\eta_{\ell}$ as a stopping criterion.

In the finite element method, the approximation spaces are built from meshes
and the Adapt step usually consists in local mesh-refinement. Since the
spatial domain associated with the mathematical problem is subdivided into
cells $K$, say triangles in two dimensions, the solution $u_{N}$ is obtained
by collecting over all the local cell contributions $u_{K}$, representing the
approximation of the solution $u$ on the cell $K$:

$u_{K}=u_{\ell}|_{K}.$ (12)

In order to be useful for local mesh refinement, the estimator $\eta_{\ell}$
is supposed to have a similar local structure, i.e., it is the sum of local
contributions $\eta_{K}$ corresponding to $K$:

$\eta_{\ell}=\left(\sum_{K}\eta_{K}^{{2}}\right)^{{\frac{1}{2}}}.$ (13)

The information encoded in the estimator map $\eta_{K}$ is used in different
ways to adapt or refine the mesh. The main idea is to split locally a cell
$K$, if the local error indicator $\eta_{K}$ is considered too large, see for
instance Pelle et al. (1996) or Dörfler (1996). In a similar way one could use
local coarsening, i.e., neighbouring cells are glued together to define a
bigger cell where the local error indicator is small enough. The algorithm
used to select cells for refinement (or coarsening) is called Mark and the
step of mesh modification Adapt. The first natural question is about
convergence of the adaptive algorithm (which does not follow from a priori
error analysis). Under the condition of convergence, the second question is
about the speed of convergence, measured in terms of number of unknowns.

Let us sketch a short argument for convergence. We consider the step from
$\ell$ to $\ell+1$. For the sake of simplicity, we suppose that the boundary
value problem (1) corresponds to the minimisation of a quadratic energy
functional. Moreover, we suppose that the finite element spaces $V_{\ell}$
form a sequence of conforming nested spaces ($V_{\ell}\subset
V_{\ell+1}\subset V$ ) and that there holds a Galerkin orthogonality
$(e_{\ell+1},u_{\ell+1}-u_{\ell})_{V}=0$, with $(\cdot,\cdot)_{V}$ the inner
product in $V$. Then the errors $e_{\ell}:=\|u-u_{\ell}\|_{V}$ are related by

$e_{\ell+1}^{2}=e_{\ell}^{2}-\|u_{\ell+1}-u_{\ell}\|_{V}^{2}$

and we get geometrical convergence if for some constant $\beta>0$

$e_{\ell}\leq\beta\|u_{\ell+1}-u_{\ell}\|_{V}.$ (14)

since then $e_{\ell+1}\leq\rho e_{\ell}$ with $\rho=\sqrt{1-1/\beta^{2}}$.

In order to establish (14), it is convenient to assume the local lower bound

$\eta_{K}\leq c\|u_{\ell+1}-u_{\ell}\|_{K}$ (15)

for all cells $K$ from the set of refined cells
$\mathcal{R}_{\ell}\subset\mathcal{K}_{\ell}$ and to ensure that, for some
$\theta>0$, the estimator linked to the cells in $\mathcal{R}_{\ell}$ is a
least equal to a proportion $\theta$ of the total estimator:

$\sum_{K\in\mathcal{R}_{\ell}}\eta_{K}^{2}\geq\theta^{2}\sum_{K\in\mathcal{K}_{\ell}}\eta_{K}^{2}.$
(16)

Indeed, using (6) and (13), then (16) and finally (15), we get

$e_{\ell}^{2}\leq
C^{2}\sum_{K\in\mathcal{K}_{\ell}}\eta_{K}^{2}\leq\frac{C^{2}}{\theta^{2}}\sum_{K\in\mathcal{R}_{\ell}}\eta_{K}^{2}\leq\frac{C^{2}c^{2}}{\theta^{2}}\sum_{K\in\mathcal{R}_{\ell}}\|u_{\ell+1}-u_{\ell}\|_{K}^{2},$

which implies (14) with $\beta=Cc/\theta.$

The assumed bound (15) looks similar to what is called _local efficiency_ :
there exists a constant $c>0$ independent of $N$, $u_{N}$ and $K$ such that,
for every mesh cell $K$, there holds:

$\eta_{K}\leq c\|u-u_{N}\|_{\omega_{K}},$ (17)

where $\omega_{K}$ is a collection of neighbouring cells of $K$ (patch) and
$\|\cdot\|_{\omega_{K}}$ is an appropriate norm on $\omega_{K}$, see, e.g.,
Verfürth (2013). This property strictly translates the fact that if the local
discretization error is small, then the local estimator must also be small.
However, (17) does not imply (15) (the difficulty is not the presence of
$\omega_{K}$ instead of $K$), and even (17) might not always hold. It might be
thought that if an estimator does not satisfy this local efficiency property,
mesh refinement could be misled. Indeed, regions where the discretization
error is low could nevertheless be refined, as the local $\eta_{K}$ would be
free to overestimate the actual discretization error. However, it turns out in
practice that an estimator with poor efficiency can still be useful to drive
the mesh refinement algorithm (see the discussion in Carstensen and Merdon
(2010)), and can even lead to optimal meshes.

Condition (16) states that a minimum percentage of the estimator contributions
should give rise to refinement. It is usually called Dörfler marking or bulk
chasing. Clearly, (16) allows for a large number of possibilities for
selecting $\mathcal{R}_{\ell}$ and one naturally tends to refine the cells
with the largest contributions. This choice is related to our second question
about the speed of convergence. Choosing the cells to be refined as the
largest contributors allows to bound the number of additional cells by the
difference in estimators. This idea basically leads to an estimate of the
number of cells needed to produce a given accuracy, see Stevenson (2007) for
the first proof.

The unavailability of lower bounds has led to the development of several more
involved techniques, see section 3.

###  2.5 The goal-oriented paradigm

Many practical situations are focused on the approximation of a single
quantity $J$, such as the drag- and lift-coefficients, average of heat
transfer along a part of the boundary, stress intensity factors, a local
stress or strain in elasticity, etc. However, we observe that the
aforementioned theory heavily relies on the functional analytic setting and
the corresponding norms are used to measure the error $u-u_{N}$. Instead one
is interested in

$|J(u)-J(u_{N})|$

and the objective of goal-oriented error estimation is to provide estimators
for this quantity. The main idea is to compute an approximation to the
solution $z$ to a _dual_ problem that involves $J$ in its right-hand side.
This dual solution then allows to evaluate locally the sensitivity of $J$ with
respect to the discretization error leading to an error estimator in the usual
form, see for instance Becker and Rannacher (2001), and also Becker and
Rannacher (1996); Giles and Süli (2002); González-Estrada et al. (2014); Han
(2005); Prudhomme and Oden (1999); Rognes and Logg (2013).

##  3 Theory of adaptive finite element methods

The mathematical theory of adaptive finite elements started with the work of
Ivo Babuška in the 1970s (see Babuška and Rheinboldt (1978)), and found
increasing interest with the works of Claes Johnson (see Eriksson and Johnson
(1985)), Mark Ainsworth and John Tinsley Oden (see Ainsworth and Oden (1993,
2000)), Pedro Morin, Ricardo Nochetto and Kunibert G. Siebert (see Morin et
al. (2000)), and Rüdiger Verfürth (see Verfürth (1999)) in the 1990s. Since
then, it has made important theoretical progress, that we summarise below.

###  3.1 Optimality of adaptive algorithms

New insights have been brought to the theory of adaptive algorithms. We first
define what is meant by the notion of optimality. We assume a given procedure
that, for any mesh $\mathcal{K}$, computes an approximation $u_{\mathcal{K}}$
to the solution $u$ of a given partial differential equation, usually by a
finite element method based on approximation spaces $V_{\mathcal{K}}$ whose
dimensions are denoted $N(\mathcal{K})$. We further assume possible to
approximate the solution $u$ by elements of $V_{\mathcal{K}}$ at a certain
convergence rate $s>0$, which mathematically translates into: there exists an
absolute constant $C>0$ such that

$\inf_{v_{\mathcal{K}}\in V_{\mathcal{K}},N(\mathcal{K})\leq
N}\left\|u-v_{\mathcal{K}}\right\|\leq CN^{-s}\quad\forall N\in\mathbb{N},$

in an appropriate norm. Then, an adaptive algorithm constructing a (in
principle infinite) sequence of meshes
$(\mathcal{K}_{\ell})_{\ell=1,2,\cdots}$ and consequent solutions
$(u_{\ell})_{\ell=1,2,\cdots}$ is said to be (quasi-)optimal, if the sequence
$(u_{\ell})_{\ell=1,2,\cdots}$ has a similar error decay:

$\left\|u-u_{\ell}\right\|\leq
C\left(N(\mathcal{K}_{\ell})\right)^{-s}\quad\forall\ell\in\mathbb{N},$

(with possibly a different constant $C$). One should mention that such an
optimality result is much stronger than the classical _a priori_ finite
element theory, since no direct assumption on the regularity of the solution
is made.

This theory brings the adaptive finite element technology close to the
nonlinear approximation theory of multi-resolution wavelet methods, see, e.g.,
Cohen et al. (2001) and Karaivanov and Petrushev (2003). It has been a very
active research field, starting with the works of Dörfler (1996); Morin et al.
(2000). The combination with nonlinear approximation theory has been achieved
in Binev et al. (2004); Stevenson (2007), which lead to many further results
for different methods/equations/algorithms, see for example Cascon et al.
(2008); Becker and Mao (2008); Bonito and Nochetto (2010); Ferraz-Leite et al.
(2010); Kreuzer and Georgoulis (2018); Buffa et al. (2022). Now it has
attained a state of maturity in the context of second-order linear elliptic
partial differential equations, certain finite element methods and mesh
refinement algorithms. The appearance of an underlying structure of the
available results has lead to the Axiom-paper of Carstensen et al. (2014).
Despite the large number of articles about application of goal-oriented
adaptivity, theoretical results concerning their optimality are still sparse.
Mommer and Stevenson (2009) provides the first optimality result, using the
product of primal and dual estimators as an upper bound to the functional
error; see also Becker et al. (2011); Feischl et al. (2016).

###  3.2 Other recent achievements

Let us mention first that the described theory of adaptive finite elements has
been generalised to nonlinear monotone equations by Feischl et al. (2014) and
saddle-point systems as the Stokes equations, see Becker and Mao (2011);
Bringmann and Carstensen (2017); Feischl (2019, 2022). Also progress has been
made in the understanding of parabolic problems, see Ern et al. (2017).

Moreover, the theory has been generalised to control the overall work of the
adaptive algorithm, and not just the dimension of the discrete spaces, see,
e.g., Haberl et al. (2021); Becker et al. (2023); Févotte et al. (2024). This
is particularly important from a practical viewpoint, since the various
stopping criteria in the different nested loops can then be harmonised.

##  4 Current engineering practice(s)

Now let us turn to the practical side of numerical methods, when they are used
in practical computations in physics, applied sciences and industrial
applications. In this context, there have already been some initiatives to
incorporate error control based on a posteriori error estimators as well as
adaptive discretization techniques, see for instance the special volume Plewa
et al. (2005) published earlier. Nevertheless, the true potential of all these
techniques seems to remain largely untapped.

###  4.1 Accuracy vs. computational cost vs. complexity

For numerical simulation oriented towards real applications, such as in the
technological and industrial sector, the challenges are, at least, threefold:

  1. 1.

Simulation of complex systems, that may encompass, for instance, complex
three-dimensional geometries, nonlinear partial differential equations with
nonlinear boundary conditions, multi-physics or multi-scale phenomena, complex
materials, etc.

  2. 2.

Simulation with reduced computation time and/or limited computational
resources may be of major importance for certain applications, with the
possible goal to even provide predictions in real-time.

  3. 3.

Certified numerical simulation, with trustworthy and accurate predictions.

It is clear that the third point is related to error control and adaptive
discretizations. In fact, the notions of verification and validation of finite
element simulations have always been of importance within the numerical
simulation community, see, e.g., Babuska and Oden (2004); Babuška and Oden
(2006); Babuška et al. (2007); Stein (2014); Szabó and Babuška (2021). However
this may be balanced or even be in conflict with the first two points, and
this partly explains why error control is sometimes an undertaken issue,
because it is not the top one priority for some practitioners devoted to a
specific goal. Our personal experience is that precision is often sacrificed
for speed. One of our goals in these volumes is to show that these two
requirements can be reconciled by the adaptivity and optimality provided by a
posteriori error estimation.

###  4.2 Some initiatives

Let us however mention a few examples in which a posteriori error estimations
or control have been considered in an industrial context:

  1. 1.

For the neutron diffusion equation in nuclear reactor cores, see Ciarlet et
al. (2023) (and see also Conjungo Taumhas, Y. et al. (2023) for an evaluation
of the modelling error).

  2. 2.

For turbulent hydraulic and thermal-hydraulic simulations with a perspective
of subsequent applications to the nuclear energy sector, see Nassreddine et
al. (2022, 2023) and Dakroub et al. (2023).

  3. 3.

Still with applications inspired by the nuclear industry, Liu et al. (2017)
address contact problems in elastostatics.

  4. 4.

For real-time surgical simulation, see Bui et al. (2018b). as well as more
details in section 5.1 below.

  5. 5.

Still for perspectives in surgical simulation and modelling of soft tissue,
see Duprez et al. (2020) and Bui et al. (2023).

  6. 6.

In the aeronautical sector, viscous compressible flows around airfoils were
investigated in Basile et al. (2022)

Remark that these works are a step toward certified numerical simulation and
have been highly motivated by a context where safety issues are fundamental.
Of course, while there are not many existing works, the above list is not
exhaustive, and other illustrations are highlighted in some of the
contributions to these volumes.

###  4.3 Some bottlenecks

According to our own experience, we can point out the following bottlenecks
relative to error control and adaptive discretization in routine industrial
computations:

  1. 1.

For trustworthy certified numerical simulation, not only the discretization
errors need to be controlled, but also modelling errors and uncertainties in
parameter calibration. These latter can be of larger magnitude and it can be
more involved to evaluate and reduce them.

  2. 2.

The design of a posteriori error estimators is, to a large extent, specific to
the model under consideration, and it is challenging and time-consuming to
design a posteriori error estimators well-suited for a complex model related
to an industrial application.

  3. 3.

The efficient implementation of some error estimation techniques is not always
straightforward in industrial or commercial codes.

  4. 4.

Error control and adaptive techniques are not always part of training related
to numerical simulation, and a large part of the literature on the topic is
not easily accessible to outsiders.

Note finally that part of the current activity in the field is inspired from
these bottlenecks. Just to mention an example, various recent works have been
dedicated to explore the interplay between adaptive finite element meshing and
uncertainty quantification technologies, see Bespalov et al. (2022); Bespalov
and Silvester (2023); Eigel and Merdon (2016); Eigel et al. (2016); Guignard
et al. (2016); Guignard and Nobile (2018); Oden et al. (2005).

##  5 Special topics and industrial applications

Error estimation for numerical methods plays an important role in quality
assurance. It enables users to better control the quality of their
simulations. So-called non-standard numerical methods have also seen some
advances in terms of error estimation. We will mention here a few numerical
methods which offer alternatives to the standard finite element method based
on Lagrange polynomials (see 2.2). We also discuss some of the most novel
practical applications where a posteriori error estimates have been
introduced, including surgical simulation and industrial scale fracture
mechanics.

###  5.1 Real-time/interactive simulations

Because of the increasing need for interactive simulations, both for medical
applications and robotic control and computer animations, but also in order to
build digital twins of actual systems, the last 10 years or so have seen major
progress towards real-time quality control of simulated solutions.

This poses a number of problems which will be discussed in forthcoming special
issues of Advances in Applied Mechanics. The first difficulty lies in
computing the error estimate fast enough for the solution to be usable within
the constraints of the practical application. For example, in surgical
simulation, surgical guidance and surgical training, a range from 50 (for
visual feedback) to 500 (for haptic feedback) solutions per second are
necessary. Moreover, it is necessary to refine the mesh locally, based on the
error estimate, within the time constraints posed by the application at hand.
The interested reader can refer to the work of Bui et al. (2018a, b, 2019),
which show the first real-time error estimation techniques for interactive
mechanics simulations of bodies undergoing large deformations.

Real-time error estimation for interactive deformable bodies still poses
several challenges, especially as the demand for more realistic simulations in
applications such as virtual surgery, computer animation, and virtual reality
continues to grow. Here are some key challenges:

Computational Efficiency

Developing error estimation techniques that can operate within the constraints
of real-time simulation environments is crucial. These methods need to be
computationally efficient to provide accurate estimates without significantly
impacting the overall simulation performance.

Integration with Simulation Frameworks

Integrating error estimation techniques seamlessly into existing simulation
frameworks poses a challenge. These techniques need to be compatible with
various simulation algorithms, such as finite element, meshfree, or position-
based methods, to ensure widespread adoption across different applications.

Sensitivity to Simulation Parameters

Error estimation methods should be robust and reliable across different
simulation scenarios and parameter settings. They should account for
uncertainties and variations in material properties, boundary conditions, and
external forces to provide accurate estimates in diverse environments. In the
context of biomechanics, recent progress was made in a series of papers which
attempt to disentangle model error from discretization error Hauseux et al.
(2017a, b, 2018). In general, this connection between model and discretization
error seems one of the richest direction of investigation, in particular when
real-time data acquisition is possible and in the context of machine learning
and artificial intelligence surrogate acceleration methods, e.g. Deshpande et
al. (2022, 2023).

User Interaction and Control

Providing intuitive interfaces for users to interactively control error
estimation and refinement processes is essential. Users should have the
flexibility to adjust parameters, set error tolerances, and guide adaptive
refinement strategies in real time to achieve desired simulation outcomes. A
closer interaction between the human user and the computer could allow to
switch between total computer control and total human control depending on the
error level.

Guaranteed upper bounds

In real-time simulations of surgery, guaranteed upper bounds are crucial for
ensuring the safety and effectiveness of the procedures. These bounds provide
maximum limits on various parameters such as response time, computational
complexity, and error rates. For instance, in robotic surgery, having
guaranteed upper bounds on latency ensures that movements are executed within
acceptable time frames to prevent accidents or errors. Similarly,
computational complexity bounds ensure that simulations run efficiently
without overwhelming hardware resources, allowing for smooth and accurate
real-time feedback during surgical procedures. Overall, these guaranteed upper
bounds are essential for maintaining reliability, accuracy, and safety in
real-life applications like surgery simulations.

In short, one of the key remaining questions could be stated as: _“Given
multi-modal experimental observations obtained, in the best case, in real-
time, and an underlying model, how sufficient is the data acquired to simulate
the system ? How important and uncertain is the form of the model ? How
important are the parameters used ? Are we better off simulating more
scenarios (offline or online) or should we make more measurements ?”_. This
direction seems ripe for fruitful investigations, as summarised in a recent
review Eftimie et al. (2023).

###  5.2 Strain smoothing

Smoothed finite element methods see e.g. a review Nguyen-Xuan et al. (2008a)
are based on the idea of transforming derivatives of shape functions into
products with outward normals of smoothing cells. This suppresses the need to
use Jacobian transformations, enables polytopal meshes, makes it possible to
compute on extremely distorted meshes, alleviates numerical locking and can,
in certain conditions provide upper bounds for the system’s energy. Several
approaches were developed, mostly based on the methods shown in Nguyen-Xuan et
al. (2008a).

Little work has been done in the area of error estimates for strain smoothing
stabilized finite elements. The main question lies in the choice of the number
of subcells. One subcell may provide equivalent methods to stress-based finite
elements (dual), giving an upper bound to the energy. In the infinite limit,
as the number of subcells goes to infinity, the standard displacement based
(primal) finite element method is recovered. González-Estrada et al. (2013)
presents ideas on how to obtain error indicators for smoothed finite element
methods.

###  5.3 Partition of unity enrichment

Extended finite element methods are partition of unity methods, similar to the
generalized finite element method and a few others, including hp-clouds. The
idea of the method Bordas and Menk (2023)) is to enrich the approximation by
special functions which introduce known features of the exact solution. The
idea is to improve the approximation property of the finite element (or
meshfree) space by making it possible to reproduce the known features of the
solution. See the book Bordas and Menk (2023) and the recent review on error
estimates for enriched approximations in González-Estrada et al. (2023).

For example, to model a crack, a discontinuous function is introduced to take
into account the jump across the crack surface. Asymptotic functions can be
introduced to improve the accuracy of the solution close to the crack front.
These methods were used in a wide variety of contexts, and called for specific
treatments in terms of error estimation. Those are summarized in Bordas and
Menk (2023). They were also implemented in a commercial code MorfeoCrack,
based on the developments in the following seminal paper Jin et al. (2017).

In short, standard recovery based methods fail because they are based on
smoothing (projection on a polynomial space), whilst the enrichment functions
are usually non-polynomial, see, e.g., Bordas and Duflot (2007). This leads to
smearing of the recovered solution, defeating its purpose to provide a higher
quality solution to compare the raw finite element counterpart. Residual based
methods have also to be adapted because of the extra terms present in the
approximations, see Gerasimov et al. (2012); Hild et al. (2009); Rüter et al.
(2013).

Several approaches exist for this, including subtracting the enrichment, see
Ródenas et al. (2008), or projecting on enriched spaces which are able to take
into account the special features present in the solution space, see Bordas et
al. (2008); Bordas and Duflot (2007); Duflot and Bordas (2008). Other
approaches were shown in the literature, such as Panetier et al. (2010);
Prange et al. (2012); Rüter et al. (2013) and, recently, in Bento et al.
(2023), higher-order methods were investigated.

The goal-oriented techniques mentioned in 2.5 can be extended, and a number of
contributions exist in the field of goal-oriented error estimation for
enriched approximations. The interested reader can refer to González-Estrada
et al. (2013, 2014, 2021, 2023).

The above papers introduce goal-oriented error estimates (GOEE), which
quantify and control local errors in quantities of interest (QoI) for advanced
engineering applications, such as aerospace, see also Section 2.5. This
includes a recovery-based error estimation technique for QoIs is presented,
using an enhanced version of the Superconvergent Patch Recovery (SPR)
technique, see Zienkiewicz and Zhu (1987). This approach provides nearly
statically admissible stress fields, resulting in accurate estimations of
local discretization error contributions to QoI. The technique requires
reasonable computational cost and could be easily implemented into finite
element codes or used as an independent postprocessing tool.

The error estimation in QoI relies on evaluating QoI through solving auxiliary
problems. Energy estimates are used to relate errors in QoI to the initial
problem and auxiliary problem solutions. Explicit and implicit residual-based
approaches and smoothing techniques for energy estimates can be used, as well
as the Zienkiewicz-Zhu estimate and SPR techniques. Bridging these approaches
aims to obtain guaranteed upper bounds while retaining ease of implementation.
The SPR-CX approach is an efficient and simple goal-oriented adaptivity
procedure for linear QoI in elasticity problems, extended to handle singular
elasticity problems.

###  5.4 How important is it that the recovered (smoothed) solution satisfy
the boundary conditions and the governing equations?

We discuss here briefly the question of statical admissibility of recovered
solutions, presented in the following: González-Estrada et al. (2012); Bordas
and Menk (2023); González-Estrada et al. (2023). These contributions assesses
the effect of statical admissibility and the ability of recovered solutions to
represent singular solutions, along with the accuracy, local, and global
effectivity of recovery-based error estimators for enriched finite element
methods, such as the extended finite element method (XFEM). Two recovery
techniques are studied: the superconvergent patch recovery procedure with
equilibration and enrichment (SPR-CX) and the extended moving least squares
recovery (XMLS). Both techniques enrich the recovered solutions, with SPR-CX
enforcing equilibrium constraints.

_Numerical results highlight the necessity of extended recovery techniques in
error estimators for this class of problems, with statically admissible
recovered solutions yielding significant improvements in effectivities. This
emphasizes the importance of both extended recovery procedures and statical
admissibility for accurate assessment of the quality of enriched finite
element approximations._

###  5.5 Meshfree/meshless methods

Meshfree methods are numerical techniques used for solving partial
differential equations without relying on a predefined mesh. Unlike
traditional methods like finite element analysis, meshfree methods operate
directly on scattered data points, making them particularly useful for
problems with complex geometries or evolving domains. One notable advantage is
their flexibility in handling irregular geometries and dynamic problems, which
often pose challenges for mesh-based approaches.

A significant strength of meshfree methods lies in their ability to reproduce
complex phenomena with high fidelity, especially in situations where
traditional mesh-based methods struggle due to mesh distortion or excessive
refinement requirements. Additionally, meshfree methods often exhibit good
scalability and efficiency, particularly for problems involving large
deformations or adaptive refinement.

However, these methods also have their limitations. Reproducibility can be
challenging since the results depend on the distribution of nodes or data
points, which may vary between different simulations. Furthermore, achieving
smooth solutions can be difficult, especially in regions with sparse data or
irregular distributions of nodes.

For a comprehensive understanding of meshfree methods, one can refer to review
papers such as Nguyen et al. (2008), which provide an in-depth analysis of
various aspects, including the theoretical foundations, implementation
strategies, and applications of meshfree methods.

When it comes to error estimation in meshfree methods, several factors need to
be taken into account. These include the choice of basis functions or shape
functions, the accuracy of the numerical integration scheme, and the
interpolation error associated with the scattered data points. Proper error
estimation is crucial for assessing the reliability and accuracy of the
computed solutions and guiding the refinement or adaptation strategies.
Importantly, certain meshfree methods suffer from conditioning issues which
may lead to algebraic errors, as discussed in this volume of Advances in
Applied Mechanics, see Papež (2024). In particular this can apply to methods
based on collocation, i.e., methods that, conversely to Galerkin techniques
presented in 2.2, consist in writing the strong form directly at “quasi-
arbitrary” sets of nodes within the domain, see Jacquemin et al. (2023) for a
review of adaptive schemes in meshless collocation, also known as the smart
cloud method.

It is important to note that in meshfree methods, Galerkin orthogonality, a
property commonly satisfied in traditional finite element methods, may not
hold on the boundary. This is because the test functions employed in meshfree
formulations typically do not vanish on the boundary, leading to deviations
from orthogonality. Understanding and managing such deviations are essential
for ensuring the accuracy and stability of the numerical solutions,
particularly near domain boundaries.

Additionally, meshless methods do not employ polynomial approximations, which
makes Gauss quadrature inexact. It is therefore critical to contain
integration errors. This can be done by an interplay between using more
background subcells in regions of interest as well as increasing the number of
integration points per subcell, see Racz and Bui (2012).

The interested reader can refer to the following papers for additional
references on error analysis for meshfree methods: Arnold and Wendland (1983);
Davydov and Oanh (2011); Dolbow and Belytschko (1998); Li et al. (2020);
Orkisz and Milewski (2008); Park et al. (2003); Perazzo et al. (2008); Rabczuk
and Belytschko (2005), which includes applications to shell elements, optimal
point placement in collocation methods, a posteriori error estimation driving
point cloud adaptation, and applications to localized phenomena and large
gradients.

###  5.6 Isogeometric analysis

Isogeometric analysis (IGA) is a computational technique that integrates the
geometric design and analysis of structures or materials. Unlike traditional
finite element methods, IGA employs the same basis functions to represent both
the geometry and the solution field, typically using Non-Uniform Rational
B-Splines (NURBS) or other spline-based representations. This seamless
integration of geometric and simulation aspects offers several advantages over
traditional methods.

One of the key strengths of IGA lies in its ability to precisely represent
complex geometries using the same basis functions employed for the analysis,
thereby eliminating the need for mesh generation and simplifying the workflow.
This leads to significant reductions in pre-processing time and enables more
accurate representations of geometric features.

Moreover, IGA facilitates the use of higher-order basis functions, which can
lead to more accurate solutions, particularly for problems involving curved or
irregular boundaries. By leveraging the smoothness properties of spline
functions, IGA often produces smoother and more realistic solutions compared
to traditional finite element methods.

However, despite these advantages, IGA also has its challenges. One notable
limitation is the computational cost associated with the construction and
manipulation of spline representations, especially for problems involving
large-scale simulations or dynamic analyses. Additionally, the coupling
between the geometry and analysis introduces complexities in the formulation
and implementation of boundary conditions and geometric modifications, see,
e.g., Hu et al. (2018); Nguyen et al. (2014).

For a deeper understanding of IGA, interested individuals can refer to review
papers such as Nguyen et al. (2015), which provide comprehensive insights into
the theoretical foundations, computational aspects, and applications of
isogeometric analysis.

When it comes to error estimation in IGA, similar considerations apply as in
traditional finite element methods, including the choice of basis functions,
integration schemes, and interpolation errors. Proper error estimation is
essential for assessing the accuracy and reliability of the computed solutions
and guiding refinement strategies.

Isogeometric analysis offers a powerful framework for integrating geometric
design and analysis, with advantages in accuracy, efficiency, and geometric
flexibility. However, challenges remain in terms of computational cost and the
formulation of boundary conditions, highlighting the ongoing research efforts
in this field.

The key difficulties in adaptive isogeometric simulations lie in the tensor-
product nature of the underlying shape functions (NURBS), which, if nothing is
done, leads to spurious propagation of refinements throughout the domain.
Alternatives such as the geometry independent field approximation techniques
avoid such issues by enabling the use of local refinement (through splines
such as PHT splines, for example), whilst keeping the geometrical
representation of the domain identical, and approximated by NURBS, see
Atroshchenko et al. (2018); Jansari et al. (2022a); Videla et al. (2019,
2024).

Other exciting directions of research are provided in the following recent
contributions including space-time analysis, as well as functional-type error
estimates for isogeometric analysis, see Langer et al. (2016); Matculevich
(2018); Langer et al. (2019).

##  6 Perspectives

In Advances in Applied Mechanics (AAMS) Vol 58: Error control, adaptive
discretizations, and applications, Part 1, world-leading authors present
cutting-edge research at the intersection of computational mechanics and
applied mathematics, exploring innovative approaches to error control and
adaptive discretizations across various fields.

## Chapters

  1. 1.

$hp$ adaptive Discontinuous Galerkin strategies driven by a posteriori error
estimation with application to aeronautical flow problems (Chapelier et al.,
2024) presents recent developments about $h$, $p$, and $hp$-adaptive
strategies driven by a posteriori error estimators using a high-order
Discontinuous Galerkin finite element numerical framework, where $h$ stands
for the mesh size and $p$ for the polynomial degree of the finite element
approximation. A combination of error estimators tailored for the numerical
method considered is presented along with smoothness indicators driving the
decision to refine in $h$ or $p$. The $h$, $p$, and $hp$-adaptation strategies
are described and applied to flow problems of aeronautical interest, including
scale-resolving simulations of the transitional flow past a NACA0012 airfoil,
scale-resolving simulations of the turbulent jet issued by a realistic nozzle
geometry, and inviscid simulations of the transonic flow around a complex CRM
aircraft geometry. For all cases considered, the interest of $h$, $p$, and
$hp$-adaptation is demonstrated for easing the meshing process and increasing
the resolution in flow regions of interest, enabling a significant reduction
of the total number of computational degrees of freedom compared to manual
meshing techniques and classical lower-order numerical approximations.

  2. 2.

An anisotropic mesh adaptation method based on gradient recovery and optimal
shape elements (Fortin, 2024) The authors present a complete mesh adaptation
strategy applicable for controlling the discretization error on a finite
element solution of Lagrange type of any degree. The method is described in
detail, outlining the process of constructing a more accurate solution from a
finite element solution using a gradient recovery method. The error is then
estimated as the difference between the reconstructed and the initial
solution. Subsequently, the mesh is modified using local operations to
minimize the error on the gradient. Additionally, a number of numerical
examples are provided to illustrate the effectiveness of the approach.

  3. 3.

Model reduction techniques for parametrized nonlinear partial differential
equations (Nguyen, 2024) The authors present model reduction techniques for
parametrized nonlinear partial differential equations (PDEs). The main
ingredients of their approach include reduced basis (RB) spaces to provide
rapidly convergent approximations to the parametric manifold, Galerkin
projection of the underlying PDEs onto the RB space to reduce the number of
degrees of freedom, and empirical interpolation schemes to rapidly evaluate
the nonlinear terms associated with the Galerkin projection. They devise a
first-order empirical interpolation method to construct an inexpensive and
stable interpolation of the nonlinear terms. Two different hyper-reduction
strategies are considered: hyper-reduction followed by linearization, and
linearization followed by hyper-reduction. The authors extend empirical
interpolation to nonintrusive model reduction and apply it to compressible
flows in both supersonic and hypersonic regimes. They present numerical
results to illustrate the accuracy, efficiency, and stability of the reduced-
order models. The interested reader can refer to Kerfriden et al. (2014);
Hoang et al. (2016); Goury et al. (2016); Agathos et al. (2020); Hoang et al.
(2021, 2022); Chen et al. (2024) for adaptive (certified) model order
reduction for non-linear problems and localised phenomena such as fracture,
molecular dynamics and inverse problems.

  4. 4.

Adaptive finite elements for obstacle problems (Gustafsson, 2024) The author
summarize three applications of the obstacle problem to membrane contact,
elastoplastic torsion, and cavitation modelling, demonstrating how the
resulting models can be solved using mixed finite elements. He highlights the
challenge of constructing fixed computational meshes for any inequality-
constrained problem due to the unknown shape of the coincidence set.
Consequently, he demonstrates how $h$–adaptivity can be utilized to resolve
the unknown coincidence set. Additionally, the author discusses practical
challenges that must be overcome in the application of the adaptive methods.

  5. 5.

A Posteriori Error Identities and Estimates of modelling Errors (Repin, 2024)
This Chapter discusses a posteriori estimation methods for mathematical models
based on partial differential equations. The analysis is based on functional
identities of a special kind, reflecting the most general relations that hold
for deviations from the exact solution of a boundary value problem. These
identities do not depend on special properties of approximations and contain
no mesh-dependent constants, making them valid for any function in the
admissible (energy) class. This universality enables the control of accuracy
for various numerical approximations and the comparison of solutions of
mathematical models. The capabilities are demonstrated with the paradigm of
modelling errors generated by simplifications of the original problem. Three
groups of problems are discussed, where errors of simplification have
different origins: errors arising from simplifying coefficients of the
equation, errors associated with simplifying geometry, and dimension reduction
errors. It is shown that in any of these cases, the desired error estimates
follow from the general a posteriori identity after proper specification of
the functional spaces and operators associated with the boundary value
problem.

  6. 6.

Exact error control for variational problems via convex duality and explicit
flux reconstruction (Bartels and Kaltenbach, 2024) A posteriori error
estimates serve as crucial tools for bounding discretization errors in terms
of computable quantities, bypassing the need for establishing often
challenging regularity conditions. However, for problems involving non-
linearity, non-differentiability, jumping coefficients, or finite element
methods with anisotropic triangulations, such estimates can often result in
large factors, leading to sub-optimal error estimates. To address this issue,
exact and explicit error representations are derived using convex duality
arguments, effectively avoiding such effects.

  7. 7.

Algebraic error in numerical PDEs and its estimation (Papež, 2024) The paper
discusses error estimation for the Poisson model problem, focusing on
residual-based and flux reconstruction error estimates. Residual-based
estimates, while useful for estimating discretization errors, have limitations
due to the lack of exact algebraic solutions. The paper proposes adjustments
to these estimates but highlights drawbacks. Flux reconstruction offers local
error indicators and guaranteed bounds but is computationally intensive. The
paper emphasizes the importance of reducing algebraic error for accurate
discretization error estimation and suggests further research on adaptive mesh
refinement and algebraic solver stopping criteria. It also advocates for
rigorous methods to mitigate algebraic errors in various numerical methods. It
is worth noting that other methods including meshfree methods, enriched
partition of unity methods, see Agathos et al. (2016b, a, 2018, 2019b, 2019a,
2020); Bordas and Menk (2023), isogeometric analysis, see Langer et al.
(2016); Yu et al. (2018); Matculevich (2018); Videla et al. (2019); Langer et
al. (2019); Jansari et al. (2022b) and collocation methods, see Arnold and
Wendland (1983); Jia et al. (2019), smart point clouds, see Perazzo et al.
(2008); Jacquemin et al. (2023), are also concerned by algebraic errors due to
ill-conditioning of the stiffness matrix. The interested reader can refer to
the following papers.

## References

  * Agathos et al. (2019a) Agathos, K., Bordas, S.P., Chatzi, E., 2019a.  Improving the conditioning of xfem/gfem for fracture mechanics problems through enrichment quasi-orthogonalization.  Computer Methods in Applied Mechanics and Engineering 346, 1051–1073.
  * Agathos et al. (2020) Agathos, K., Bordas, S.P., Chatzi, E., 2020.  Parametrized reduced order modeling for cracked solids.  International Journal for Numerical Methods in Engineering 121, 4537–4565.
  * Agathos et al. (2016a) Agathos, K., Chatzi, E., Bordas, S.P., 2016a.  Stable 3d extended finite elements with higher order enrichment for accurate non planar fracture.  Computer Methods in Applied Mechanics and Engineering 306, 19–46.
  * Agathos et al. (2019b) Agathos, K., Chatzi, E., Bordas, S.P., 2019b.  A unified enrichment approach addressing blending and conditioning issues in enriched finite elements.  Computer Methods in Applied Mechanics and Engineering 349, 673–700.
  * Agathos et al. (2016b) Agathos, K., Chatzi, E., Bordas, S.P., Talaslidis, D., 2016b.  A well-conditioned and optimally convergent xfem for 3d linear elastic fracture.  International Journal for Numerical Methods in Engineering 105, 643–677.
  * Agathos et al. (2018) Agathos, K., Ventura, G., Chatzi, E., Bordas, S.P., 2018\.  Stable 3d xfem/vector level sets for non-planar 3d crack propagation and comparison of enrichment schemes.  International Journal for Numerical Methods in Engineering 113, 252–276.
  * Ainsworth and Oden (1993) Ainsworth, M., Oden, J.T., 1993\.  A unified approach to a posteriori error estimation using element residual methods.  Numerische Mathematik 65, 23–50.
  * Ainsworth and Oden (2000) Ainsworth, M., Oden, J.T., 2000\.  A posteriori error estimation in finite element analysis.  Pure Appl. Math., Wiley-Intersci. Ser. Texts Monogr. Tracts, Chichester: Wiley.
  * Arnold (1982) Arnold, D.N., 1982.  An interior penalty finite element method with discontinuous elements.  SIAM Journal on Numerical Analysis 19, 742–760.
  * Arnold and Wendland (1983) Arnold, D.N., Wendland, W.L., 1983\.  On the asymptotic convergence of collocation methods.  Mathematics of Computation 41, 349–381.
  * Atroshchenko et al. (2018) Atroshchenko, E., Tomar, S., Xu, G., Bordas, S.P.A., 2018\.  Weakening the tight coupling between geometry and simulation in isogeometric analysis: from sub- and super-geometric analysis to geometry-independent field approximation (GIFT).  International Journal for Numerical Methods in Engineering 114, 1131–1159.
  * Babuška et al. (2007) Babuška, I., Nobile, F., Tempone, R., 2007.  Reliability of computational science.  Numerical Methods for Partial Differential Equations 23, 753–784.
  * Babuska and Oden (2004) Babuska, I., Oden, J.T., 2004\.  Verification and validation in computational engineering and science: basic concepts.  Computer Methods in Applied Mechanics and Engineering 193, 4057–4066.
  * Babuška and Oden (2006) Babuška, I., Oden, J.T., 2006\.  The reliability of computer predictions: can they be trusted?  International Journal of Numerical Analysis and Modeling 3, 255–272.
  * Babuška and Rheinboldt (1978) Babuška, I., Rheinboldt, W.C., 1978\.  Error estimates for adaptive finite element computations.  SIAM Journal on Numerical Analysis 15, 736–754.
  * Babuška et al. (1992) Babuška, I., Durán, R., Rodríguez, R., 1992.  Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements.  SIAM Journal on Numerical Analysis 29, 947–964.
  * Bank and Weiser (1985) Bank, R.E., Weiser, A., 1985\.  Some a posteriori error estimators for elliptic partial differential equations.  Mathematics of Computation 44, 283–301.
  * Bartels and Kaltenbach (2024) Bartels, S., Kaltenbach, A., 2024\.  Exact error control for variational problems via convex duality and explicit flux reconstruction.  Advances in Applied Mechanics (AAMS) 58\.
  * Basile et al. (2022) Basile, F., Chapelier, J.B., de la Llave Plata, M., Laraufie, R., Frey, P., 2022.  Unstructured $h$\- and _hp_ -adaptive strategies for discontinuous galerkin methods based on _a posteriori_ error estimation for compressible flows.  Comput. Fluids 233, 21\.  doi:10.1016/j.compfluid.2021.105245. id/No 105245\.
  * Becker et al. (2023) Becker, R., Brunner, M., Innerberger, M., Melenk, J.M., Praetorius, D., 2023.  Cost-optimal adaptive iterative linearized FEM for semilinear elliptic PDEs.  European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 57, 2193–2225.
  * Becker et al. (2011) Becker, R., Estecahandy, E., Trujillo, D., 2011.  Weighted marking for goal-oriented adaptive finite element methods.  SIAM Journal on Numerical Analysis 49, 2451–2469.
  * Becker and Mao (2008) Becker, R., Mao, S., 2008.  An optimally convergent adaptive mixed finite element method.  Numerische Mathematik 111, 35–54.
  * Becker and Mao (2011) Becker, R., Mao, S., 2011.  Quasi-optimality of adaptive non-conforming finite element methods for the Stokes equations.  SIAM Journal on Numerical Analysis 49, 970–991.
  * Becker and Rannacher (1996) Becker, R., Rannacher, R., 1996\.  A feed-back approach to error control in finite element methods: Basic analysis and examples.  East-West Journal of Numerical Mathematics 4, 237–264.
  * Becker and Rannacher (2001) Becker, R., Rannacher, R., 2001\.  An optimal control approach to a posteriori error estimation in finite element methods.  Acta Numerica 10, 1–102.
  * Beirão da Veiga et al. (2013) Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A., 2013\.  Basic principles of virtual element methods.  Mathematical Models and Methods in Applied Sciences 23, 199–214.
  * Bento et al. (2023) Bento, M.H., Proença, S.P., Duarte, C.A., 2023.  Recovery strategies, a posteriori error estimation, and local error indication for second-order g/xfem and fem.  International Journal for Numerical Methods in Engineering 124, 3025–3062.
  * Bernardi and Maday (1997) Bernardi, C., Maday, Y., 1997\.  Spectral methods, in: Handbook of numerical analysis, Vol. V. North-Holland, Amsterdam. volume V of Handb. Numer. Anal., pp. 209–485.
  * Bertoluzza (1995) Bertoluzza, S., 1995.  A posteriori error estimates for the wavelet Galerkin method.  Applied Mathematics Letters 8, 1–6.
  * Bertoluzza et al. (1994) Bertoluzza, S., Naldi, G., Ravel, J.C., 1994.  Wavelet methods for the numerical solution of boundary value problems on the interval, in: Wavelets: theory, algorithms, and applications. Proceedings of the international conference on wavelets, held in Taormina, Italy, October 14-20, 1993. San Diego, CA: Academic Press, pp. 425–448.
  * Bespalov and Silvester (2023) Bespalov, A., Silvester, D., 2023\.  Error estimation and adaptivity for stochastic collocation finite elements Part II: Multilevel approximation.  SIAM Journal on Scientific Computing 45, A784–A800.
  * Bespalov et al. (2022) Bespalov, A., Silvester, D.J., Xu, F., 2022.  Error estimation and adaptivity for stochastic collocation finite elements Part I: Single-level approximation.  SIAM Journal on Scientific Computing 44, A3393–A3412.
  * Binev et al. (2004) Binev, P., Dahmen, W., DeVore, R., 2004.  Adaptive finite element methods with convergence rates.  Numerische Mathematik 97, 219–268.
  * Bonito and Nochetto (2010) Bonito, A., Nochetto, R.H., 2010\.  Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method.  SIAM Journal on Numerical Analysis 48, 734–771.
  * Bordas and Duflot (2007) Bordas, S., Duflot, M., 2007\.  Derivative recovery and a posteriori error estimate for extended finite elements.  Computer Methods in Applied Mechanics and Engineering 196, 3381–3399.
  * Bordas et al. (2008) Bordas, S., Duflot, M., Le, P., 2008.  A simple error estimator for extended finite elements.  Communications in Numerical Methods in Engineering 24, 961–971.
  * Bordas and Menk (2023) Bordas, S., Menk, A., 2023.  Partition of Unity Methods.  Wiley.
  * Braess and Schöberl (2008) Braess, D., Schöberl, J., 2008\.  Equilibrated residual error estimator for edge elements.  Mathematics of Computation 77, 651–672.
  * Brenner and Scott (2008) Brenner, S.C., Scott, L.R., 2008\.  The mathematical theory of finite element methods. volume 15 of Texts in Applied Mathematics.  Third ed., Springer, New York.
  * Bringmann and Carstensen (2017) Bringmann, P., Carstensen, C., 2017\.  An adaptive least-squares FEM for the Stokes equations with optimal convergence rates.  Numerische Mathematik 135, 459–492.
  * Buffa et al. (2022) Buffa, A., Gantner, G., Giannelli, C., Praetorius, D., Vázquez, R., 2022.  Mathematical foundations of adaptive isogeometric analysis.  Archives of Computational Methods in Engineering 29, 4479–4555.
  * Bui et al. (2023) Bui, H.P., Duprez, M., Rohan, P.Y., Lejeune, A., Bordas, S.P.A., Bucki, M., Chouly, F., 2023.  Automatic mesh refinement for soft tissue.  URL: https://hal.science/hal-04208610. hAL preprint.
  * Bui et al. (2019) Bui, H.P., Tomar, S., Bordas, S.P., 2019.  Corotational cut finite element method for real-time surgical simulation: Application to needle insertion simulation.  Computer Methods in Applied Mechanics and Engineering 345, 183–211.
  * Bui et al. (2018a) Bui, H.P., Tomar, S., Courtecuisse, H., Audette, M., Cotin, S., Bordas, S.P., 2018a.  Controlling the error on target motion through real-time mesh adaptation: applications to deep brain stimulation.  International Journal for Numerical Methods in Biomedical Engineering 34, e2958.
  * Bui et al. (2018b) Bui, H.P., Tomar, S., Courtecuisse, H., Cotin, S., Bordas, S.P.A., 2018b.  Real-time error control for surgical simulation.  IEEE Transactions on Biomedical Engineering 65, 596–607.
  * Burman et al. (2015) Burman, E., Claus, S., Hansbo, P., Larson, M.G., Massing, A., 2015.  CutFEM: discretizing geometry and partial differential equations.  International Journal for Numerical Methods in Engineering 104, 472–501.
  * Canuto et al. (2006) Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A., 2006\.  Spectral methods.  Scientific Computation, Springer-Verlag, Berlin.
  * Carstensen et al. (2014) Carstensen, C., Feischl, M., Page, M., Praetorius, D., 2014\.  Axioms of adaptivity.  Computers & Mathematics with Applications 67, 1195–1253.
  * Carstensen and Merdon (2010) Carstensen, C., Merdon, C., 2010\.  Estimator competition for Poisson problems.  Journal of Computational Mathematics 28, 309–330.
  * Cascon et al. (2008) Cascon, J.M., Kreuzer, C., Nochetto, R.H., Siebert, K.G., 2008\.  Quasi-optimal convergence rate for an adaptive finite element method.  SIAM Journal on Numerical Analysis 46, 2524–2550.
  * Chamoin and Legoll (2023) Chamoin, L., Legoll, F., 2023\.  An introductory review on a posteriori error estimation in finite element computations.  SIAM Review 65, 963–1028.
  * Chapelier et al. (2024) Chapelier, J.B., Basile, F., Naddei, F., Plata, M.D.L.L., Couaillier, V., Laraufie, R., 2024\.  hp adaptive discontinuous galerkin strategies driven by a posteriori error estimation with application to aeronautical flow problems.  Advances in Applied Mechanics (AAMS) 58\.
  * Chen et al. (2024) Chen, L., Wang, Z., Lian, H., Ma, Y., Meng, Z., Li, P., Ding, C., Bordas, S.P., 2024.  Reduced order isogeometric boundary element methods for cad-integrated shape optimization in electromagnetic scattering.  Computer Methods in Applied Mechanics and Engineering 419, 116654.
  * Ciarlet et al. (2023) Ciarlet, P., Do, M.H., Madiot, F., 2023.  A _posteriori_ error estimates for mixed finite element discretizations of the neutron diffusion equations.  European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 57, 1–27.
  * Ciarlet (2002) Ciarlet, P.G., 2002.  The finite element method for elliptic problems. volume 40 of Classics in Applied Mathematics.  Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  * Cicuttin et al. (2021) Cicuttin, M., Ern, A., Pignet, N., 2021.  Hybrid high-order methods—a primer with applications to solid mechanics.  SpringerBriefs in Mathematics, Springer, Cham.
  * Cockburn et al. (2016) Cockburn, B., Di Pietro, D.A., Ern, A., 2016.  Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods.  ESAIM. Mathematical Modelling and Numerical Analysis 50, 635–650.
  * Cohen et al. (2001) Cohen, A., Dahmen, W., DeVore, R., 2001.  Adaptive wavelet methods for elliptic operator equations: convergence rates.  Mathematics of Computation 70, 27–75.
  * Cohen and Masson (2000) Cohen, A., Masson, R., 2000\.  Wavelet adaptive method for second order elliptic problems: Boundary conditions and domain decomposition.  Numerische Mathematik 86, 193–238.
  * Conjungo Taumhas, Y. et al. (2023) Conjungo Taumhas, Y. , Labeurthre, D., Madiot, F., Mula, O., Taddei, T., 2023.  Impact of physical model error on state estimation for neutronics applications*.  ESAIM: Proceedings 73, 158–172.
  * Cottrell et al. (2009) Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y., 2009.  Isogeometric analysis.  John Wiley & Sons, Ltd., Chichester.
  * Dakroub et al. (2023) Dakroub, J., Faddoul, J., Omnes, P., Sayah, T., 2023\.  A posteriori error estimates for the time-dependent Navier-Stokes system coupled with the convection-diffusion-reaction equation.  Advances in Computational Mathematics 49, 60.  Id/No 67.
  * Davydov and Oanh (2011) Davydov, O., Oanh, D.T., 2011\.  Adaptive meshless centres and rbf stencils for poisson equation.  Journal of Computational Physics 230, 287–304.
  * Demkowicz and Gopalakrishnan (2010) Demkowicz, L., Gopalakrishnan, J., 2010\.  A class of discontinuous Petrov-Galerkin methods. Part I: the transport equation.  Computer Methods in Applied Mechanics and Engineering 199, 1558–1572.
  * Demkowicz et al. (2012) Demkowicz, L., Gopalakrishnan, J., Niemi, A.H., 2012.  A class of discontinuous Petrov-Galerkin methods. Part III: Adaptivity.  Applied Numerical Mathematics 62, 396–427.
  * Deshpande et al. (2022) Deshpande, S., Lengiewicz, J., Bordas, S.P., 2022.  Probabilistic deep learning for real-time large deformation simulations.  Computer Methods in Applied Mechanics and Engineering 398, 115307.
  * Deshpande et al. (2023) Deshpande, S., Sosa, R.I., Bordas, S., Lengiewicz, J., 2023\.  Convolution, aggregation and attention based deep neural networks for accelerating simulations in mechanics.  Frontiers in Materials 10, 1128954\.
  * Di Pietro and Droniou (2020) Di Pietro, D.A., Droniou, J., 2020\.  The hybrid high-order method for polytopal meshes. volume 19 of MS&A. Modeling, Simulation and Applications.  Springer, Cham.
  * Di Pietro and Ern (2012) Di Pietro, D.A., Ern, A., 2012\.  Mathematical aspects of discontinuous Galerkin methods. volume 69 of Mathematics & Applications.  Springer, Heidelberg.
  * Dolbow and Belytschko (1998) Dolbow, J., Belytschko, T., 1998\.  An introduction to programming the meshless Element FreeGalerkin method.  Archives of Computational Methods in Engineering 5, 207–241.
  * Dong and Ern (2022) Dong, Z., Ern, A., 2022.  Hybrid high-order and weak Galerkin methods for the biharmonic problem.  SIAM Journal on Numerical Analysis 60, 2626–2656.
  * Dörfler (1996) Dörfler, W., 1996.  A convergent adaptive algorithm for Poisson’s equation.  SIAM Journal on Numerical Analysis 33, 1106–1124.
  * Duflot and Bordas (2008) Duflot, M., Bordas, S., 2008\.  A posteriori error estimation for extended finite elements by an extended global recovery.  International Journal for Numerical Methods in Engineering 76, 1123–1138.
  * Duprez et al. (2020) Duprez, M., Bordas, S.P.A., Bucki, M., Bui, H.P., Chouly, F., Lleras, V., Lobos, C., Lozinski, A., Rohan, P.Y., Tomar, S., 2020\.  Quantifying discretization errors for soft tissue simulation in computer assisted surgery: a preliminary study.  Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems 77, 709–723.
  * Duprez and Lozinski (2020) Duprez, M., Lozinski, A., 2020\.  $\phi$-FEM: a finite element method on domains defined by level-sets.  SIAM Journal on Numerical Analysis 58, 1008–1028.
  * Durán et al. (1991) Durán, R., Muschietti, M.A., Rodríguez, R., 1991.  On the asymptotic exactness of error estimators for linear triangular finite elements.  Numerische Mathematik 59, 107–127.
  * Eftimie et al. (2023) Eftimie, R., Mavrodin, A., Bordas, S.P., 2023.  From digital control to digital twins in medicine: A brief review and future perspectives.  Advances in Applied Mechanics 56, 323–368.
  * Eigel and Merdon (2016) Eigel, M., Merdon, C., 2016\.  Local equilibration error estimators for guaranteed error control in adaptive stochastic higher-order Galerkin finite element methods.  SIAM/ASA Journal on Uncertainty Quantification 4, 1372–1397.
  * Eigel et al. (2016) Eigel, M., Merdon, C., Neumann, J., 2016.  An adaptive multilevel Monte Carlo method with stochastic bounds for quantities of interest with uncertain data.  SIAM/ASA Journal on Uncertainty Quantification 4, 1219–1245.
  * Eriksson and Johnson (1985) Eriksson, K., Johnson, C., 1985\.  Error estimates and automatic time step control for non-linear parabolic problems. I.  Bericht. Universität Jyväskylä. Mathematisches Institut 31.
  * Ern and Guermond (2004) Ern, A., Guermond, J.L., 2004\.  Theory and practice of finite elements. volume 159 of Applied Mathematical Sciences.  Springer-Verlag, New York.
  * Ern and Guermond (2021) Ern, A., Guermond, J.L., 2021\.  Finite elements. I—Approximation and interpolation. volume 72 of Texts in Applied Mathematics.  Springer, Cham.
  * Ern et al. (2017) Ern, A., Smears, I., Vohralík, M., 2017.  Guaranteed, locally space-time efficient, and polynomial-degree robust a posteriori error estimates for high-order discretizations of parabolic problems.  SIAM Journal on Numerical Analysis 55, 2811–2834.
  * Ern and Vohralík (2013) Ern, A., Vohralík, M., 2013\.  Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs.  SIAM Journal on Scientific Computing 35, A1761–A1791.
  * Feischl (2019) Feischl, M., 2019.  Optimality of a standard adaptive finite element method for the Stokes problem.  SIAM Journal on Numerical Analysis 57, 1124–1157.
  * Feischl (2022) Feischl, M., 2022.  Inf-sup stability implies quasi-orthogonality.  Mathematics of Computation 91, 2059–2094.
  * Feischl et al. (2014) Feischl, M., Führer, T., Praetorius, D., 2014.  Adaptive FEM with optimal convergence rates for a certain class of nonsymmetric and possibly nonlinear problems.  SIAM Journal on Numerical Analysis 52, 601–625.
  * Feischl et al. (2016) Feischl, M., Praetorius, D., van der Zee, K.G., 2016.  An abstract analysis of optimal goal-oriented adaptivity.  SIAM Journal on Numerical Analysis 54, 1423–1448.
  * Ferraz-Leite et al. (2010) Ferraz-Leite, S., Ortner, C., Praetorius, D., 2010.  Convergence of simple adaptive Galerkin schemes based on $h-h/2$ error estimators.  Numerische Mathematik 116, 291–316.
  * Févotte et al. (2024) Févotte, F., Rappaport, A., Vohralík, M., 2024.  Adaptive regularization, discretization, and linearization for nonsmooth problems based on primal-dual gap estimators.  Computer Methods in Applied Mechanics and Engineering 418, Paper No. 116558, 33.
  * Fortin (2024) Fortin, A., 2024.  An anisotropic mesh adaptation method based on gradient recovery and optimal shape elements.  Advances in Applied Mechanics (AAMS) 58\.
  * Gander and Wanner (2012) Gander, M.J., Wanner, G., 2012\.  From Euler, Ritz, and Galerkin to modern computing.  SIAM Review 54, 627–666.
  * Gerasimov et al. (2012) Gerasimov, T., Rüter, M., Stein, E., 2012.  An explicit residual-type error estimator for Q1-quadrilateral extended finite element method in two-dimensional linear elastic fracture mechanics.  International Journal for Numerical Methods in Engineering 90, 1118–1155.
  * Giles and Süli (2002) Giles, M.B., Süli, E., 2002\.  Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality.  Acta Numerica 11, 145–236.
  * Glowinski et al. (1994) Glowinski, R., Pan, T., Périaux, J., 1994.  A fictitious domain method for Dirichlet problem and applications.  Computer Methods in Applied Mechanics and Engineering 111, 283–303.
  * González-Estrada et al. (2014) González-Estrada, O.A., Nadal, E., Ródenas, J.J., Kerfriden, P., Bordas, S.P.A., Fuenmayor, F.J., 2014\.  Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery.  Computational Mechanics 53, 957–976.
  * González-Estrada et al. (2021) González-Estrada, O.A., Natarajan, S., Ródenas, J.J., Bordas, S.P., 2021\.  Error estimation for the polygonal finite element method for smooth and singular linear elasticity.  Computers & Mathematics with Applications 92, 109–119.
  * González-Estrada et al. (2013) González-Estrada, O.A., Natarajan, S., Ródenas, J.J., Nguyen-Xuan, H., Bordas, S.P., 2013.  Efficient recovery-based error estimation for the smoothed finite element method for smooth and singular linear elasticity.  Computational Mechanics 52, 37–52.
  * González-Estrada et al. (2012) González-Estrada, O.A., Ródenas, J.J., Bordas, S.P.A., Duflot, M., Kerfriden, P., Giner, E., 2012\.  On the role of enrichment and statical admissibility of recovered fields in a posteriori error estimation for enriched finite element methods.  Engineering Computations 29, 814–841.
  * González-Estrada et al. (2023) González-Estrada, O.A., Ródenas García, J.J., Bordas, S.P., 2023.  Recovery-based error estimation and bounding in xfem.  Partition of Unity Methods , 145–190.
  * Goury et al. (2016) Goury, O., Amsallem, D., Bordas, S.P.A., Liu, W.K., Kerfriden, P., 2016.  Automatised selection of load paths to construct reduced-order models in computational damage micromechanics: from dissipation-driven random selection to bayesian optimization.  Computational Mechanics 58, 213–234.
  * Grepl et al. (2007) Grepl, M.A., Maday, Y., Nguyen, N.C., Patera, A.T., 2007\.  Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations.  M2AN. Mathematical Modelling and Numerical Analysis 41, 575–605.
  * Guignard and Nobile (2018) Guignard, D., Nobile, F., 2018\.  A posteriori error estimation for the stochastic collocation finite element method.  SIAM Journal on Numerical Analysis 56, 3121–3143.
  * Guignard et al. (2016) Guignard, D., Nobile, F., Picasso, M., 2016.  A posteriori error estimation for elliptic partial differential equations with small uncertainties.  Numerical Methods for Partial Differential Equations. An International Journal 32, 175–212.
  * Gustafsson (2024) Gustafsson, T., 2024.  Adaptive finite elements for obstacle problems.  Advances in Applied Mechanics (AAMS) 58\.
  * Haberl et al. (2021) Haberl, A., Praetorius, D., Schimanko, S., Vohralík, M., 2021\.  Convergence and quasi-optimal cost of adaptive algorithms for nonlinear operators including iterative linearization and algebraic solver.  Numerische Mathematik 147, 679–725.
  * Han (2005) Han, W., 2005.  A posteriori error analysis via duality theory. With applications in modeling and numerical approximations. volume 8 of Advances in Mechanics and Mathematics.  New York, NY: Springer.
  * Haslinger and Renard (2009) Haslinger, J., Renard, Y., 2009\.  A new fictitious domain approach inspired by the extended finite element method.  SIAM Journal on Numerical Analysis 47, 1474–1499.
  * Hauseux et al. (2017a) Hauseux, P., Hale, J.S., Bordas, S.P., 2017a.  Accelerating monte carlo estimation with derivatives of high-level finite element models.  Computer Methods in Applied Mechanics and Engineering 318, 917–936.
  * Hauseux et al. (2017b) Hauseux, P., Hale, J.S., Bordas, S.P., 2017b.  Calculating the malliavin derivative of some stochastic mechanics problems.  Plos one 12, e0189994.
  * Hauseux et al. (2018) Hauseux, P., Hale, J.S., Cotin, S., Bordas, S.P., 2018\.  Quantifying the uncertainty in a hyperelastic soft tissue model with stochastic parameters.  Applied Mathematical Modelling 62, 86–102.
  * Hesthaven et al. (2016) Hesthaven, J.S., Rozza, G., Stamm, B., 2016.  Certified reduced basis methods for parametrized partial differential equations.  SpringerBriefs in Mathematics, Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao.  BCAM SpringerBriefs.
  * Hild et al. (2009) Hild, P., Lleras, V., Renard, Y., 2009.  A posteriori error analysis for Poisson’s equation approximated by XFEM.  European Series in Applied and Industrial Mathematics (ESAIM): Proceedings 27, 107–121.
  * Hoang et al. (2021) Hoang, C., Choi, Y., Carlberg, K., 2021.  Domain-decomposition least-squares Petrov–Galerkin (DD-LSPG) nonlinear model reduction.  Computer Methods in Applied Mechanics and Engineering 384, 113997.
  * Hoang et al. (2022) Hoang, C., Chowdhary, K., Lee, K., Ray, J., 2022.  Projection-based model reduction of dynamical systems using space–time subspace and machine learning.  Computer Methods in Applied Mechanics and Engineering 389, 114341.
  * Hoang et al. (2016) Hoang, K., Kerfriden, P., Bordas, S.P.A., 2016.  A fast, certified and “tuning free” two-field reduced basis method for the metamodelling of affinely-parametrised elasticity problems.  Computer Methods in Applied Mechanics and Engineering 298, 121–158.
  * Hu et al. (2018) Hu, Q., Chouly, F., Hu, P., Cheng, G., Bordas, S.P.A., 2018\.  Skew-symmetric Nitsche’s formulation in isogeometric analysis: Dirichlet and symmetry conditions, patch coupling and frictionless contact.  Computer Methods in Applied Mechanics and Engineering 341, 188–220.
  * Jacquemin et al. (2023) Jacquemin, T., Suchde, P., Bordas, S.P., 2023.  Smart cloud collocation: geometry-aware adaptivity directly from cad.  Computer-Aided Design 154, 103409\.
  * Jansari et al. (2022a) Jansari, C., Videla, J., Natarajan, S., Bordas, S.P., Atroshchenko, E., 2022a.  Adaptive enriched geometry independent field approximation for 2d time-harmonic acoustics.  Computers & Structures 263, 106728\.
  * Jansari et al. (2022b) Jansari, C., Videla, J., Natarajan, S., Bordas, S.P., Atroshchenko, E., 2022b.  Adaptive enriched geometry independent field approximation for 2d time-harmonic acoustics.  Computers & Structures 263, 106728\.
  * Jia et al. (2019) Jia, Y., Anitescu, C., Zhang, Y.J., Rabczuk, T., 2019\.  An adaptive isogeometric analysis collocation method with a recovery-based error estimator.  Computer Methods in Applied Mechanics and Engineering 345, 52–74.
  * Jin et al. (2017) Jin, Y., González-Estrada, O., Pierard, O., Bordas, S., 2017\.  Error-controlled adaptive extended finite element method for 3d linear elastic crack propagation.  Computer Methods in Applied Mechanics and Engineering 318, 319–348.
  * Karaivanov and Petrushev (2003) Karaivanov, B., Petrushev, P., 2003\.  Nonlinear piecewise polynomial approximation beyond Besov spaces.  Applied and Computational Harmonic Analysis 15, 177–223.
  * Kerfriden et al. (2011) Kerfriden, P., Gosselet, P., Adhikari, S., Bordas, S.P.A., 2011\.  Bridging proper orthogonal decomposition methods and augmented Newton-Krylov algorithms: an adaptive model order reduction for highly nonlinear mechanical problems.  Computer Methods in Applied Mechanics and Engineering 200, 850–866.
  * Kerfriden et al. (2014) Kerfriden, P., Ródenas, J.J., Bordas, S.A., 2014.  Certification of projection-based reduced order modelling in computational homogenisation by the constitutive relation error.  International Journal for Numerical Methods in Engineering 97, 395–422.
  * Kreuzer and Georgoulis (2018) Kreuzer, C., Georgoulis, E.H., 2018\.  Convergence of adaptive discontinuous Galerkin methods.  Mathematics of Computation 87, 2611–2640.
  * Ladeveze and Leguillon (1983) Ladeveze, P., Leguillon, D., 1983\.  Error estimate procedure in the finite element method and applications.  SIAM Journal on Numerical Analysis 20, 485–509.
  * Langer et al. (2016) Langer, U., Matculevich, S., Repin, S., 2016.  A posteriori error estimates for space-time iga approximations to parabolic initial boundary value problems.  arXiv preprint arXiv:1612.08998 .
  * Langer et al. (2019) Langer, U., Matculevich, S., Repin, S., 2019.  Guaranteed error bounds and local indicators for adaptive solvers using stabilised space–time iga approximations to parabolic problems.  Computers & Mathematics with Applications 78, 2641–2671.
  * Lemaire (2021) Lemaire, S., 2021.  Bridging the hybrid high-order and virtual element methods.  IMA Journal of Numerical Analysis 41, 549–593.
  * Lesaint and Raviart (1974) Lesaint, P., Raviart, P.A., 1974\.  On a finite element method for solving the neutron transport equation, in: Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974). Academic Press, New York-London, pp. 89–123.
  * Li et al. (2020) Li, W., Nguyen-Thanh, N., Huang, J., Zhou, K., 2020\.  Adaptive analysis of crack propagation in thin-shell structures via an isogeometric-meshfree moving least-squares approach.  Computer Methods in Applied Mechanics and Engineering 358, 112613.
  * Liu et al. (2007) Liu, G.R., Dai, K.Y., Nguyen, T.T., 2007.  A smoothed finite element method for mechanics problems.  Computational Mechanics 39, 859–877.
  * Liu et al. (2017) Liu, H., Ramière, I., Lebon, F., 2017.  On the coupling of local multilevel mesh refinement and ZZ methods for unilateral frictional contact problems in elastostatics.  Computer Methods in Applied Mechanics and Engineering 323, 1–26.
  * Lozinski (2019) Lozinski, A., 2019.  A primal discontinuous Galerkin method with static condensation on very general meshes.  Numerische Mathematik 143, 583–604.
  * Matculevich (2018) Matculevich, S., 2018.  Functional approach to the error control in adaptive iga schemes for elliptic boundary value problems.  Journal of Computational and Applied Mathematics 344, 394–423.
  * Moës et al. (2006) Moës, N., Béchet, E., Tourbier, M., 2006.  Imposing Dirichlet boundary conditions in the extended finite element method.  International Journal for Numerical Methods in Engineering 67, 1641–1669.
  * Mommer and Stevenson (2009) Mommer, M.S., Stevenson, R., 2009\.  A goal-oriented adaptive finite element method with convergence rates.  SIAM Journal on Numerical Analysis 47, 861–886.
  * Monasse and Perrier (1998) Monasse, P., Perrier, V., 1998\.  Orthonormal wavelet bases adapted for partial differential equations with boundary conditions.  SIAM Journal on Mathematical Analysis 29, 1040–1065.
  * Morin et al. (2000) Morin, P., Nochetto, R.H., Siebert, K.G., 2000.  Data oscillation and convergence of adaptive FEM.  SIAM Journal on Numerical Analysis 38, 466–488.
  * Nassreddine et al. (2022) Nassreddine, G., Omnes, P., Sayah, T., 2022.  A posteriori error estimates for the large eddy simulation applied to stationary Navier-Stokes equations.  Numerical Methods for Partial Differential Equations 38, 1468–1498.
  * Nassreddine et al. (2023) Nassreddine, G., Omnes, P., Sayah, T., 2023.  _A posteriori_ error estimates for the large eddy simulation applied to incompressible fluids.  European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 57, 2159–2191.
  * Neittaanmäki and Repin (2004) Neittaanmäki, P., Repin, S., 2004\.  Reliable methods for computer simulation. volume 33 of Studies in Mathematics and its Applications.  Elsevier Science B.V., Amsterdam.
  * Nguyen (2024) Nguyen, N.C., 2024.  Model reduction techniques for parametrized nonlinear partial differential equations.  Advances in Applied Mechanics (AAMS) 58\.
  * Nguyen et al. (2015) Nguyen, V.P., Anitescu, C., Bordas, S.P., Rabczuk, T., 2015\.  Isogeometric analysis: an overview and computer implementation aspects.  Mathematics and Computers in Simulation 117, 89–116.
  * Nguyen et al. (2014) Nguyen, V.P., Kerfriden, P., Brino, M., Bordas, S.P.A., Bonisoli, E., 2014.  Nitsche’s method for two and three dimensional NURBS patch coupling.  Computational Mechanics 53, 1163–1182.
  * Nguyen et al. (2008) Nguyen, V.P., Rabczuk, T., Bordas, S.P.A., Duflot, M., 2008\.  Meshless methods: a review and computer implementation aspects.  Mathematics and Computers in Simulation 79, 763–813.
  * Nguyen-Xuan et al. (2008a) Nguyen-Xuan, H., Bordas, S., Nguyen-Dang, H., 2008a.  Smooth finite element methods: convergence, accuracy and properties.  International Journal for Numerical Methods in Engineering 74, 175–208.
  * Nguyen-Xuan et al. (2008b) Nguyen-Xuan, H., Rabczuk, T., Bordas, S., Debongnie, J.F., 2008b.  A smoothed finite element method for plate analysis.  Computer Methods in Applied Mechanics and Engineering 197, 1184–1203.
  * Nochetto and Veeser (2012) Nochetto, R.H., Veeser, A., 2012\.  Primer of adaptive finite element methods, in: Multiscale and adaptivity: modeling, numerics and applications. Springer, Heidelberg. volume 2040 of Lecture Notes in Math., pp. 125–225.
  * Oden et al. (2005) Oden, J.T., Babuška, I., Nobile, F., Feng, Y., Tempone, R., 2005.  Theory and methodology for estimation and control of errors due to modeling, approximation, and uncertainty.  Computer Methods in Applied Mechanics and Engineering 194, 195–204.
  * Orkisz and Milewski (2008) Orkisz, J., Milewski, S., 2008\.  A’posteriori error estimation based on higher order approximation in the meshless finite difference method, in: Meshfree Methods for Partial Differential Equations IV, Springer. pp. 189–213.
  * Panetier et al. (2010) Panetier, J., Ladeveze, P., Chamoin, L., 2010.  Strict and effective bounds in goal-oriented error estimation applied to fracture mechanics problems solved with xfem.  International Journal for Numerical Methods in Engineering 81, 671–700.
  * Papež (2024) Papež, J., 2024.  Algebraic error in numerical pdes and its estimation.  Advances in Applied Mechanics (AAMS) 58\.
  * Park et al. (2003) Park, S.H., Kwon, K.C., Youn, S.K., 2003.  A posteriori error estimates and an adaptive scheme of least-squares meshfree method.  International Journal for Numerical Methods in Engineering 58, 1213–1250.
  * Pelle et al. (1996) Pelle, J.P., Beckers, P., Gallimard, L., 1996.  Estimation des erreurs de discrétisation et analyses adaptatives, application à l’automatisation des calculs éléments finis.  Institut pour la promotion des sciences de l’ingénieur.
  * Perazzo et al. (2008) Perazzo, F., Löhner, R., Perez-Pozo, L., 2008.  Adaptive methodology for meshless finite point method.  Advances in Engineering Software 39, 156–166.
  * Peskin (2002) Peskin, C.S., 2002.  The immersed boundary method.  Acta Numerica 11, 479–517.
  * Plewa et al. (2005) Plewa, T., Linde, T., Weirs, V.G. (Eds.), 2005.  Adaptive mesh refinement – theory and applications. Proceedings of the Chicago workshop on adaptive mesh refinement methods, Chicago, IL, USA, September 3–5, 2003.. volume 41 of Lect. Notes Comput. Sci. Eng. Berlin: Springer.
  * Prange et al. (2012) Prange, C., Loehnert, S., Wriggers, P., 2012.  Error estimation for crack simulations using the xfem.  International Journal for Numerical Methods in Engineering 91, 1459–1474.
  * Prudhomme and Oden (1999) Prudhomme, S., Oden, J.T., 1999\.  On goal-oriented error estimation for elliptic problems: Application to the control of pointwise errors.  Computer Methods in Applied Mechanics and Engineering 176, 313–331.
  * Quarteroni and Valli (1994) Quarteroni, A., Valli, A., 1994\.  Numerical approximation of partial differential equations. volume 23 of Springer Series in Computational Mathematics.  Springer-Verlag, Berlin.
  * Rabczuk and Belytschko (2005) Rabczuk, T., Belytschko, T., 2005\.  Adaptivity for structured meshfree particle methods in 2d and 3d.  International Journal for Numerical Methods in Engineering 63, 1559–1582.
  * Racz and Bui (2012) Racz, D., Bui, T.Q., 2012.  Novel adaptive meshfree integration techniques in meshless methods.  International Journal for Numerical Methods in Engineering 90, 1414–1434.
  * Repin (2024) Repin, S.I., 2024.  A posteriori error identities and estimates of modelling errors.  Advances in Applied Mechanics (AAMS) 58\.
  * Ródenas et al. (2008) Ródenas, J.J., González-Estrada, O.A., Tarancón, J.E., Fuenmayor, F.J., 2008.  A recovery-type error estimator for the extended finite element method based on singular+ smooth stress field splitting.  International Journal for Numerical Methods in Engineering 76, 545–571.
  * Rognes and Logg (2013) Rognes, M.E., Logg, A., 2013\.  Automated goal-oriented error control. I: Stationary variational problems.  SIAM Journal on Scientific Computing 35, c173–c193.
  * Rüter et al. (2013) Rüter, M., Gerasimov, T., Stein, E., 2013.  Goal-oriented explicit residual-type error estimates in XFEM.  Computational Mechanics 52, 361–376.
  * Stein (2014) Stein, E., 2014.  History of the finite element method – mathematics meets mechanics. II: Mathematical foundation of primal FEM for elastic deformations, error analysis and adaptivity, in: The history of theoretical, material and computational mechanics – mathematics meets mechanics and engineering. Selected papers based on the special sessions at the GAMM 2010, Karlsruhe, Germany, GAMM 2011, Graz, Austria, GAMM 2012, Darmstadt, Germany. Berlin: Springer, pp. 443–478.
  * Stevenson (2007) Stevenson, R., 2007.  Optimality of a standard adaptive finite element method.  Foundations of Computational Mathematics 7, 245–269.
  * Sukumar and Tabarraei (2004) Sukumar, N., Tabarraei, A., 2004\.  Conforming polygonal finite elements.  International Journal for Numerical Methods in Engineering 61, 2045–2066.
  * Szabó and Babuška (2021) Szabó, B., Babuška, I., 2021\.  Finite element analysis. Formulation, verification and validation.  Wiley Series Comput. Mech.. 2nd updated and revised edition ed., Hoboken, NJ: John Wiley & Sons.  doi:10.1002/9781119426479.
  * Veeser and Verfürth (2009) Veeser, A., Verfürth, R., 2009\.  Explicit upper bounds for dual norms of residuals.  SIAM Journal on Numerical Analysis 47, 2387–2405.
  * Veeser and Verfürth (2012) Veeser, A., Verfürth, R., 2012\.  Poincaré constants for finite element stars.  IMA Journal of Numerical Analysis 32, 30–47.
  * Verfürth (1999) Verfürth, R., 1999.  Error estimates for some quasi-interpolation operators.  M2AN. Mathematical Modelling and Numerical Analysis 33, 695–713.
  * Verfürth (2009) Verfürth, R., 2009.  A note on constant-free a posteriori error estimates.  SIAM Journal on Numerical Analysis 47, 3180–3194.
  * Verfürth (2013) Verfürth, R., 2013.  A posteriori error estimation techniques for finite element methods.  Numer. Math. Sci. Comput., Oxford: Oxford University Press.
  * Videla et al. (2019) Videla, J., Anitescu, C., Khajah, T., Bordas, S.P., Atroshchenko, E., 2019.  h-and p-adaptivity driven by recovery and residual-based error estimators for pht-splines applied to time-harmonic acoustics.  Computers & Mathematics with Applications 77, 2369–2395.
  * Videla et al. (2024) Videla, J., Shaaban, A.M., Atroshchenko, E., 2024.  Shape optimization with adaptive geometry independent field approximation (gift) in 3d time-harmonic acoustics.  Journal of Sound and Vibration , 118299\.
  * Vohralík (2007) Vohralík, M., 2007.  A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations.  SIAM Journal on Numerical Analysis 45, 1570–1599.
  * Yu et al. (2018) Yu, P., Anitescu, C., Tomar, S., Bordas, S.P.A., Kerfriden, P., 2018.  Adaptive isogeometric analysis for plate vibrations: an efficient approach of local refinement based on hierarchical a posteriori error estimation.  Computer Methods in Applied Mechanics and Engineering 342, 251–286.
  * Zienkiewicz and Zhu (1987) Zienkiewicz, O.C., Zhu, J.Z., 1987\.  A simple error estimator and adaptive procedure for practical engineering analysis.  International Journal for Numerical Methods in Engineering 24, 337–357.