# Lower eigenvalue bounds with hybrid high-order methods

Ngoc Tien Tran  Institut für Mathematik, Universität Augsburg, 86159 Augsburg,
Germany<EMAIL_ADDRESS>
(Date: 28th August 2024)

###### Abstract.

This paper proposes hybrid high-order eigensolvers for the computation of
guaranteed lower eigenvalue bounds. These bounds display higher order
convergence rates and are accessible to adaptive mesh-refining algorithms. The
involved constants arise from local embeddings and are available for all
polynomial degrees. A wide range of applications is possible including the
linear elasticity and Steklov eigenvalue problem.

###### Key words and phrases:

lower eigenvalue bounds, hybrid high-order

###### 1991 Mathematics Subject Classification:

65N25, 65N30

This project received funding from the European Union’s Horizon 2020 research
and innovation programme (project RandomMultiScales, grant agreement No.
865751).

##  1\. Introduction

###  1.1. Motivation

Guaranteed error control for the approximation of eigenvalues associated with
positive definite partial differential equations requires upper and lower
bounds of the exact eigenvalue $\lambda(j)$. While upper bounds can be
achieved by conforming methods using the Rayleigh-Ritz principle, lower bounds
are more challenging to obtain. The first lower eigenvalue bounds (LEB) are
computed in [22, 6] and the ideas therein have been extended to other problems
[5, 24, 21] as well. Recently, mixed methods [20] provide LEB for a large
range of examples with an additional benefit. The involved constants are
independent of the explicit discretization. The bounds of all aforementioned
methods are postprocessings of the computed eigenvalue $\lambda_{h}(j)$ of the
form

$\displaystyle\frac{\lambda_{h}(j)}{1+\delta\lambda_{h}(j)}\leq\lambda(j).$

They are restricted to lowest-order convergence because, in practice, the
parameter $\delta$ has a fixed scaling in terms of maximal mesh-size arising
from some local embeddings. Thus, higher-order methods cannot provide any
improvement and uniform mesh-refining algorithms may even outperform adaptive
computations in certain examples on nonconvex domains [9].

A remedy is [11] with a hybrid high-order (HHO) eigensolver [16, 15] plus the
Lehrenfeld-Schörberl stabilization, where the computed eigenvalue
$\lambda_{h}(j)\leq\lambda(j)$ is a direct LEB of $\lambda$ under an explicit
assumption on the maximal mesh-size. This makes the bound accessible to
higher-order discretizations and adaptivity. Other HHO schemes for the Laplace
eigenvalue problem has been considered in [4, 7] with optimal convergence
rates of adaptive mesh-refining algorithms in [8] for the lowest-order case.
The limitation of all aforementioned contributions is the restriction to the
Laplace eigenvalue problem and the involved constants are _only_ guaranteed
for lowest-order discretizations.

In this paper, we propose four abstract conditions based on the observations
of [11] that imply the bound

(1.1)
$\displaystyle\mathrm{LEB}(j)\coloneqq\min\\{1,1/(\alpha+\beta\lambda_{h}(j))\\}\lambda_{h}(j)\leq\lambda(j)$

with the convention $\mathrm{LEB}(j)\coloneqq 0$ for $\lambda_{h}(j)=\infty$.
In practice, $\alpha<1$ is a chosen parameter and $\beta$ is an explicit
constant with a positive scaling of the maximal mesh-size. Therefore,
$\lambda_{h}(j)\leq\lambda(j)$ eventually holds whenever
$\alpha+\beta\lambda_{h}(j)<1$. The main contribution is the design of HHO
methods that allow for guaranteed constants independent of the polynomial
degree. As an extra benefit, hanging nodes are allowed as long as the mesh is
partitioned into convex cells for additional flexibility in the mesh design.
Since the presented approach relies on intrinsic features of the HHO
methodology, it applies to a wide range of examples including linear
elasticity and Steklov eigenvalue problems.

###  1.2. Setting

Given a Hilbert space $V$, the eigenvalue problem of this paper seeks
eigenvalues $\lambda$ and eigenvectors $u\in V$ such that

(1.2) $\displaystyle a(u,v)=\lambda b(u,v)\quad\text{for any }v\in V.$

with a scalar product $a$ in $V$ and a symmetric positive semidefinite
bilinear form $b:V\times V\to\mathbb{R}$. They induce the norm
$\|\bullet\|_{\mathrm{a}}$ and seminorm $\|\bullet\|_{\mathrm{b}}$ in $V$. We
assume the following setting. The finite eigenvalues
$0<\lambda(1)\leq\lambda(2)\leq\dots<\infty$ are given by the Rayleigh-Ritz
principle

(1.3) $\displaystyle\lambda(j)=\min_{W\subset V,\mathrm{dim}(W)=j}\max_{v\in
W\setminus\\{0\\}}\|v\|_{a}^{2}/\|v\|_{b}^{2}.$

The corresponding eigenvectors $u(1),u(2),\dots$ are subject to the
normalization $\|u(j)\|_{b}=1$ and satisfy the orthogonality

(1.4) $\displaystyle a(u(j),u(k))=b(u(j),u(k))=0\quad\text{for any }j\neq k.$

Let $V_{\mathrm{nc}}$ be some piecewise version of $V\subset V_{\mathrm{nc}}$
with respect to a mesh, endowed with the seminorm
$\|\bullet\|_{a_{\mathrm{pw}}}$ induced by a positive semidefinite bilinear
form $a_{\mathrm{pw}}$ in $V_{\mathrm{nc}}$. It is not required that
$a_{\mathrm{pw}}$ is an extension of $a$ to $V_{\mathrm{nc}}$. On the discrete
level, $V_{h}$ denotes the discrete ansatz space with an interpolation
$\mathrm{I}_{h}:V\to V_{h}$. The prototypical HHO eigensolver seeks
eigenvalues $\lambda_{h}$ and eigenvectors $u_{h}$ such that

(1.5) $\displaystyle
a_{h}(u_{h},v_{h})+\mathrm{s}_{h}(u_{h},v_{h})=\lambda_{h}b_{h}(u_{h},v_{h})$

with symmetric positive semidefinite bilinear forms
$a_{h},b_{h},\mathrm{s}_{h}:V_{h}\times V_{h}\to\mathbb{R}$, which define the
seminorms $\|\bullet\|_{a_{h}}$, $\|\bullet\|_{b_{h}}$, and
$\|\bullet\|_{\mathrm{s}_{h}}$. In the HHO methodology, $\mathrm{s}_{h}$ is
called stabilization and $a_{h}+\mathrm{s}_{h}$ on the left-hand side of (1.5)
is a scalar product in $V_{h}$. Similar to the continuous level, the discrete
eigenvalues $0\leq\lambda_{h}(1)\leq\dots\leq\lambda_{h}(N)\leq\infty$ with
$N\coloneqq\mathrm{dim}\,V_{h}$ satisfy the Rayleigh-Ritz principle

(1.6) $\displaystyle\lambda_{h}(j)=\min_{W_{h}\subset
V_{h},\mathrm{dim}(W_{h})=j}\max_{v_{h}\in
W_{h}\setminus\\{0\\}}(\|v_{h}\|_{a_{h}}^{2}+\|v_{h}\|_{\mathrm{s}_{h}}^{2})/\|v_{h}\|_{b_{h}}^{2}.$

###  1.3. Framework

For the sake of brevity, we consider the Laplace eigenvalue problem in this
introduction with $V=H^{1}_{0}(\Omega)$,
$a\coloneqq(\nabla\bullet,\nabla\bullet)_{L^{2}(\Omega)}$,
$b\coloneqq(\bullet,\bullet)_{L^{2}(\Omega)}$, $V_{\mathrm{nc}}\coloneqq
H^{1}(\mathcal{M})$ for a mesh $\mathcal{M}$ into convex polyhedra, and
$a_{\mathrm{pw}}\coloneqq(\nabla_{\mathrm{pw}}\bullet,\nabla_{\mathrm{pw}}\bullet)_{L^{2}(\Omega)}$.
The ansatz space $V_{h}\coloneqq P_{k+1}(\mathcal{M})\times
P_{k}(\Sigma(\Omega))$ consists of piecewise polynomials on $\mathcal{M}$ and
$\Sigma(\Omega)$, the set of all interior sides, with the interpolation
$\mathrm{I}_{h}v\coloneqq(\Pi_{\mathcal{M}}^{k+1}v,\Pi_{\Sigma}^{k}v)\in
V_{h}$ for $v\in V$. Here, $\Pi_{\mathcal{M}}^{k+1}$ and $\Pi_{\Sigma}^{k}$
denote the $L^{2}$ projection onto the discrete spaces $P_{k+1}(\mathcal{M})$
and $P_{k}(\Sigma(\Omega))$. The piecewise gradient of a potential
reconstruction $\mathcal{R}_{h}:V_{h}\to W_{\mathrm{nc}}$ in the discrete
subspace $W_{\mathrm{nc}}\coloneqq P_{k+1}(\mathcal{M})$ of $V_{\mathrm{nc}}$
approximates the differential operator $\nabla$. This defines $a_{h}\coloneqq
a_{\mathrm{pw}}(\mathcal{R}_{h}\bullet,\mathcal{R}_{h}\bullet)$ and
$b_{h}(u_{h},v_{h})\coloneqq(u_{\mathcal{M}},v_{\mathcal{M}})_{L^{2}(\Omega)}$
for any
$u_{h}=(u_{\mathcal{M}},u_{\Sigma}),v_{h}=(v_{\mathcal{M}},v_{\Sigma})\in
V_{h}$. Further details on the discretization follow in Section 3 below. A
characteristic feature of the HHO methodology is the connection

(1.7)
$\displaystyle\mathcal{R}_{h}\circ\mathrm{I}_{h}=\mathrm{G}_{h}\quad\text{in
}V$

between $\mathcal{R}_{h}$ and a Galerkin projection
$\mathrm{G}_{h}:V_{\mathrm{nc}}\to W_{\mathrm{nc}}$, defined, for given $v\in
V_{\mathrm{nc}}$, as a solution to

(1.8) $\displaystyle
a_{\mathrm{pw}}(\mathrm{G}_{h}v,\varphi_{\mathrm{nc}})=a_{\mathrm{pw}}(v,\varphi_{\mathrm{nc}})\quad\text{for
any }\varphi_{\mathrm{nc}}\in W_{\mathrm{nc}}.$

Notice that $a_{\mathrm{pw}}$ is not a scalar product in $W_{\mathrm{nc}}$ and
therefore, the unique definition of $\mathrm{G}_{h}$ additionally requires
fixing the degrees of freedom associated with the kernel of the seminorm
$\|\bullet\|_{a_{\mathrm{pw}}}$ in $W_{\mathrm{nc}}$, e.g.,

(1.9)
$\displaystyle\int_{K}\mathrm{G}_{h}v\,\mathrm{d}x=\int_{K}v\,\mathrm{d}x\quad\text{for
any }K\in\mathcal{M}$

in the present model problem. From (1.8), we infer the orthogonality
$v-\mathrm{G}_{h}v\perp_{a_{\mathrm{pw}}}W_{\mathrm{nc}}$. Together with
(1.7), this shows, for any $v\in V$, that

######  (A)$a$-orthogonality

$\|\mathrm{I}_{h}v\|_{a_{h}}^{2}\leq\|v\|^{2}_{a}-\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}$

(even with equality for this example). In particular,

(1.10) $\displaystyle\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}\leq\|v\|_{a}.$

The quasi-best approximation property

######  (B)scaling of stabilization

$\|\mathrm{I}_{h}v\|_{\mathrm{s}_{h}}^{2}\leq\alpha\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}$

of the stabilization with a positive constant $\alpha$ is utilized in many
contributions [18, 2, 10]. The stabilization proposed in this paper allows for
some $\alpha$ independent of the polynomial degree $k$. The final ingredient
towards LEB involves the bound
$\|v-\mathrm{G}_{h}v\|_{L^{2}(\Omega)}^{2}\leq\beta\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}$
with $\beta\leq h_{\mathrm{max}}^{2}/\pi^{2}$ from a Poincaré inequality [1].
This, the Pythagoras theorem
$\|\mathrm{I}_{h}v\|^{2}_{b_{h}}=\|v\|_{b}^{2}-\|v-\Pi_{\mathcal{M}}^{k+1}v\|_{L^{2}(\Omega)}^{2}$,
and
$\|v-\Pi_{\mathcal{M}}^{k+1}v\|_{L^{2}(\Omega)}\leq\|v-\mathrm{G}_{h}v\|_{L^{2}(\Omega)}$
allow for

######  (C)$b$-orthogonality

$\|\mathrm{I}_{h}v\|_{b_{h}}^{2}\geq\|v\|_{b}^{2}-\beta\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}$.

Utilising similar arguments to [11], we deduce from (A)–(C) the LEB (1.1).
Since (A)–(C) are not restricted to the Laplace eigenvalue problem, the
results of this paper are applicable to other examples as well.

###  1.4. Outline

The remaining parts of this paper are organized as follows. Section 2
establishes (1.1) from (A)–(C). The remaining sections apply the results from
Section 2 to the Laplace (Section 3), Steklov (Section 4), and linear
elasticity (Section 5) eigenvalue problem. Section 6 proposes HHO eigensolvers
for the computation of guaranteed bounds of constants in local embeddings.

###  1.5. Notation

Standard notation for Lebesgue and Sobolev spaces applies throughout this
paper with the abbreviations $(\varphi,\psi)_{L^{2}(\Omega)}$ for the $L^{2}$
scalar product of two functions $\varphi,\psi\in L^{2}(\Omega)$ and
$\|\bullet\|=\|\bullet\|_{L^{2}(\Omega)}$ denotes the $L^{2}$ norm.

##  2\. Lower eigenvalue bounds

In the setting of Subsection 1.2, we recall the results from the introduction.

######  Theorem 2.1 (LEB).

Suppose that there exists an operator $\mathrm{G}_{h}:V_{\mathrm{nc}}\to
W_{\mathrm{nc}}$ onto a discrete subspace $W_{\mathrm{nc}}\subset
V_{\mathrm{nc}}$ of $V_{\mathrm{nc}}$ and positive constants $\alpha,\beta>0$
with (A)–(C) for any $v\in V$. Then (1.1) holds for any $j=1,\dots,N$.

The proof of Theorem 2.1 utilizes the following observation. Define the finite
dimensional subspace $W\coloneqq\mathrm{span}\\{u(1),\dots,u(j)\\}$ of $V$.

######  Lemma 2.2 (linear independency).

If $\beta\lambda(j)<1$, then (a)–(b) hold.

  1. (a)

The interpolations $\mathrm{I}_{h}u(1)$, $\dots$, $\mathrm{I}_{h}u(j)$ of the
exact eigenvectors $u(1),\dots,u(j)$ are linearly independent in $V_{h}$.

  2. (b)

Suppose additionally $\lambda_{h}<\infty$, then there exists $v\in W$ with
$\|v\|_{b}=1$ and

$\displaystyle(1-\alpha-\beta\lambda_{h}(j))\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}\leq\lambda(j)-\lambda_{h}(j).$

###### Proof.

The proof of Lemma 2.2 follows the arguments of [11] and is outlined below for
the sake of completeness.

##### _Proof of (a)_

Suppose otherwise, then there are real numbers $t_{1},\dots,t_{n}$ such that
$v\coloneqq t_{1}u(1)+\dots+t_{j}u(j)$ satisfies $\|v\|_{b}=1$ but
$\mathrm{I}_{h}v=0$. (The normalization $\|v\|_{b}=1$ is possible because $b$
is a norm in $\mathrm{span}\\{u(1),\dots,u(j)\\}$.) Elementary algebra with
the orthogonality in (1.4) proves
$1=\|v\|_{b}^{2}=t_{1}^{2}\|u(1)\|_{b}^{2}+\dots
t_{j}^{2}\|u(j)\|_{b}=t_{1}^{2}+\dots+t_{j}^{2}$ and therefore,

$\displaystyle\|v\|_{a}^{2}=t_{1}^{2}\|u(1)\|^{2}_{a}+\dots+t_{j}^{2}\|u(j)\|_{a}^{2}$
$\displaystyle=t_{1}^{2}\lambda(1)+\dots+t_{j}^{2}\lambda(j)$ (2.1)
$\displaystyle\leq(t_{1}^{2}+\dots+t_{j}^{2})\lambda(j)=\lambda(j).$

The combination of this with (C) and (1.10) results in

(2.2) $\displaystyle
1=\|v\|_{b}^{2}\leq\beta\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}\leq\beta\|v\|^{2}_{a}\leq\beta\lambda(j);$

a contradiction to $\beta\lambda(j)<1$.

##### _Proof of (b)_

Since the interpolations $\mathrm{I}_{h}u(1),\dots,\mathrm{I}_{h}u(j)$ are
linearly independent from (a), $\mathrm{I}_{h}W\subset V_{h}$ is a
$j$-dimensional subspace of $V_{h}$. The Rayleigh-Ritz principle (1.6) on the
discrete level shows

$\displaystyle\lambda_{h}(j)\leq\max_{v_{h}\in\mathrm{I}_{h}W\setminus\\{0\\}}(\|v_{h}\|_{a_{h}}^{2}+\|v_{h}\|_{\mathrm{s}_{h}}^{2})/\|v_{h}\|_{b_{h}}^{2}.$

Let $v\in W\setminus\\{0\\}$ with $\|v\|_{b}=1$ be given such that
$\mathrm{I}_{h}v$ is a maximizer of this Rayleigh quotient (which can attain
the value $+\infty$). Then

(2.3)
$\displaystyle\lambda_{h}(j)\|\mathrm{I}_{h}v\|_{b_{h}}^{2}\leq\|\mathrm{I}_{h}v\|_{a_{h}}^{2}+\|\mathrm{I}_{h}v\|_{\mathrm{s}_{h}}^{2}.$

From (A) and $\|v\|_{a}^{2}\leq\lambda(j)$ in (2), we deduce that
$\|\mathrm{I}_{h}v\|_{a_{h}}^{2}\leq\lambda(j)-\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}$.
This and (B) imply that the right-hand side of (2.3) is bounded by
$\lambda(j)-(1-\alpha)\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}$, while
$\lambda_{h}(j)(1-\beta\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2})$ is a
lower bound for the left-hand side of (2.3) thanks to (C) and $\|v\|_{b}=1$.
Hence,

$\displaystyle\lambda_{h}(j)(1-\beta\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2})\leq\lambda(j)-(1-\alpha)\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}.$

Reorganizing these terms concludes the proof of (b). ∎

###### Proof of Theorem 2.1.

Fix $1\leq j\leq N$. Recall the convention $\mathrm{LEB}(j)\coloneqq 0$ for
$\lambda_{h}(j)=\infty$, which is a trivial bound for $\lambda(j)$. Hence,
throughout this proof, we assume that $\lambda_{h}(j)<\infty$. The proof
consists of the two cases (a) $\alpha+\beta\lambda_{h}(j)\leq 1$ and (b)
$\alpha+\beta\lambda_{h}(j)>1$.

(a) Suppose that $\alpha+\beta\lambda_{h}(j)\leq 1$, then the assertion reads
$\lambda_{h}(j)\leq\lambda(j)$. If $\beta\lambda(j)\geq 1$, then
$\beta\lambda_{h}(j)\leq\alpha+\beta\lambda_{h}(j)\leq 1\leq\beta\lambda(j)$,
whence $\lambda_{h}(j)\leq\lambda(j)$. If $\beta\lambda(j)<1$, then Lemma
2.2(b) and $1-\alpha-\beta\lambda_{h}(j)\geq 0$ imply
$\lambda_{h}(j)\leq\lambda(j)$.

(b) Suppose that $\alpha+\beta\lambda_{h}(j)>1$, then the claim reads
$\lambda_{h}(j)/(\alpha+\beta\lambda_{h}(j))\leq\lambda(j)$. If
$\beta\lambda(j)<1$, then Lemma 2.2(b) implies the existence of a $v\in W$
with $\|v\|_{b}=1$ and
$(1-\alpha-\beta\lambda_{h}(j))\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}\leq\lambda(j)-\lambda_{h}(j)$.
The normalization $\|v\|_{b}=1$ implies $\|v\|_{a}^{2}\leq\lambda(j)$ in (2)
and so,
$\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}\leq\|v\|_{a}^{2}\leq\lambda(j)$
from (1.10). This and $1-\alpha-\beta\lambda_{h}(j)<0$ provide

$\displaystyle(1-\alpha-\beta\lambda_{h}(j))\lambda(j)\leq(1-\alpha-\beta\lambda_{h}(j))\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}\leq\lambda(j)-\lambda_{h}(j),$

whence $\lambda_{h}(j)/(\alpha+\beta\lambda_{h}(j))\leq\lambda(j)$. If
$1\leq\beta\lambda(j)$, then $\lambda_{h}(j)/(\alpha+\beta\lambda_{h}(j))\leq
1/\beta\leq\lambda(j)$. This concludes the proof. ∎

As in [11, 4, 2], the conditions (A)–(C) allow for LEB from a priori
information on the exact eigenvalue.

######  Corollary 2.3 (a priori LEB).

If $\alpha+\beta\lambda(j)\leq 1$ and $\lambda_{h}(j)<\infty$, then
$\lambda_{h}(j)\leq\lambda(j)$.

###### Proof.

The bound $\alpha+\beta\lambda(j)\leq 1$ implies
$1-\alpha-\beta\lambda_{h}(j)\geq\alpha+\beta\lambda(j)-\alpha-\beta\lambda_{h}(j)\geq\beta(\lambda(j)-\lambda_{h}(j))$.
Since $\beta\lambda(j)<1$, Lemma 2.2(b) holds and therefore,

(2.4)
$\displaystyle\beta(\lambda(j)-\lambda_{h}(j))\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}\leq\lambda(j)-\lambda_{h}(j)$

for some $v\in V$ with $\|v\|_{b}=1$. Recall
$\beta\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}\leq\beta\|v\|_{a}^{2}\leq\beta\lambda(j)$
from (1.10) and (2). This and (2.4) provide

$\displaystyle\beta\lambda(j)(\lambda(j)-\lambda_{h}(j))\leq\lambda(j)-\lambda_{h}(j).$

This inequality can only hold if $\lambda_{h}(j)\leq\lambda(j)$ because
$\beta\lambda(j)<1$. ∎

##  3\. Laplace operator

The remaining parts of this paper are devoted to the application of Theorem
2.1 to different model problems. Let $\Omega\subset\mathbb{R}^{n}$ be an open
bounded polyhedral Lipschitz domain in two or three space dimensions $n=2,3$.
The first example seeks eigenpairs $(\lambda,u)$ with

(3.1) $\displaystyle-\Delta u=\lambda u\text{ in }\Omega\quad\text{and}\quad
u=0\text{ on }\partial\Omega.$

The weak formulation of this is (1.2) with $V\coloneqq H^{1}_{0}(\Omega)$,
$a(u,v)=(\nabla u,\nabla v)_{L^{2}(\Omega)}$, and
$b(u,v)=(u,v)_{L^{2}(\Omega)}$.

###  3.1. Preliminary

Some notation on the discrete level is fixed throughout the remaining parts of
this paper before the HHO eigensolver is presented.

####  3.1.1. Triangulation

Let $\mathcal{M}$ be a finite collection of closed _convex_ nonempty polyhedra
with overlap of volume measure zero that cover
$\overline{\Omega}=\cup_{K\in\mathcal{M}}K$. A side $S$ of the mesh
$\mathcal{M}$ is a closed connected polyhedral subset of a hyperplane $H_{S}$
with positive $(n-1)$-dimensional surface measure such that either (a) there
exist $K_{+},K_{-}\in\mathcal{M}$ with $S\subset H_{S}\cap K_{+}\cap K_{-}$
(interior side) or (b) there exists $K_{+}\in\mathcal{M}$ with $S\subset
H_{S}\cap K_{+}\cap\partial\Omega$ (boundary sides). Let $\Sigma$ be a finite
collection of sides with overlap of $(n-1)$-dimensional surface measure zero
that covers the skeleton
$\partial\mathcal{M}\coloneqq\cup_{K\in\mathcal{M}}\partial
K=\cup_{S\in\Sigma}S$. The normal vector $\nu_{S}$ of an interior side
$S\in\Sigma(\Omega)$ is fixed beforehand and set $\nu_{S}\coloneqq\nu|_{S}$
for boundary sides $S\in\Sigma(\partial\Omega)$. For $S\in\Sigma(\Omega)$,
$K_{+}\in\mathcal{M}$ (resp. $K_{-}\in\mathcal{M}$) denotes the unique cell
with $S\subset\partial K_{+}$ (resp. $S\subset\partial K_{-}$) and
$\nu_{K_{+}}|_{S}=\nu_{S}$ (resp. $\nu_{K_{-}}|_{S}=-\nu_{S}$). The jump
$[v]_{S}$ of any function $v\in W^{1,1}(\mathrm{int}(T_{+}\cup
T_{-});\mathbb{R}^{m})$ along $S\in\Sigma(\Omega)$ is defined by
$[v]_{S}\coloneqq v|_{K_{+}}-v|_{K_{-}}\in L^{1}(S;\mathbb{R}^{m})$. If
$S\in\Sigma(\partial\Omega)$, then $[v]_{S}\coloneqq v|_{S}$. The set
$H^{1}(\mathcal{M})$ is the space of all piecewise $H^{1}$ function with
respect to the mesh $\mathcal{M}$. The notation $\nabla_{\mathrm{pw}}$ denotes
the piecewise application of $\nabla$ without explicit reference to
$\mathcal{M}$. This notation applies to other differential operators as well.

####  3.1.2. Trace inequality

Given a convex polyhedra $K$, let $x_{K}$ denote the midpoint of $K$. For any
side $S\in\Sigma(K)$ of $K$, $K_{S}\coloneqq\mathrm{conv}\\{x_{K},S\\}\subset
K$ is the convex hull of $S$ and $x_{K}$. We define the weight

(3.2) $\displaystyle\ell(S,K)\coloneqq|S|h_{K}^{2}/|K_{S}|.$

Notice the scaling $\ell(S,K)\approx h_{K}$ for admissible meshes (cf. [14,
Definition 1.38] for a precise definition). This paper utilizes the following
trace inequality.

######  Lemma 3.1 (trace inequality).

Any $v\in H^{1}(\mathrm{int}(K))$ with $\int_{K}v\,\mathrm{d}x=0$ satisfies

$\displaystyle\sum_{S\in\Sigma(K)}\ell(S,K)^{-1}\|v\|_{L^{2}(S)}^{2}\leq
c_{\mathrm{tr}}\|\nabla v\|_{L^{2}(K)}^{2}$

with the constant $c_{\mathrm{tr}}\coloneqq 1/\pi^{2}+2/(n\pi)$.

###### Proof.

Given $S\in\Sigma(K)$, the classical trace identity implies

(3.3)
$\displaystyle\|v\|^{2}_{L^{2}(S)}\leq\frac{|S|}{|K_{S}|}\|v\|_{L^{2}(K_{S})}^{2}+\frac{2h_{K_{S}}|S|}{n|K_{S}|}\int_{K_{S}}|v||\nabla
v|\,\mathrm{d}x$

cf., e.g., [20, proof of Lemma 7.2]. The sum of this over all $S\in\Sigma(K)$
with the notation $\ell(S,K)$ from (3.2) and $h_{K_{S}}/h_{K}\leq 1$ establish

$\displaystyle\sum_{S\in\Sigma(K)}\ell(S,K)^{-1}\|v\|_{L^{2}(S)}^{2}\leq
h_{K}^{-2}\|v\|_{L^{2}(K)}^{2}+2h_{K}^{-1}n^{-1}\int_{K}|v||\nabla
v|\,\mathrm{d}x.$

This, the Hölder, and Poincaré inequality conclude the proof. ∎

####  3.1.3. Finite element spaces

Given a subset $M\subset\mathbb{R}^{n}$ of diameter $h_{M}$, let $P_{k}(M)$
denote the space of polynomials of degree at most $k$. For any $v\in
L^{1}(M)$, $\Pi_{M}^{k}v\in P_{k}(M)$ denotes the $L^{2}$ projection of $v$
onto $P_{k}(M)$. The space of piecewise polynomials of degree at most $k$ with
respect to the mesh $\mathcal{M}$ or the sides $\Sigma$ is denoted by
$P_{k}(\mathcal{M})$ or $P_{k}(\Sigma)$. Given $v\in L^{1}(\Omega)$ (resp.
$v\in L^{1}(\partial\mathcal{M})$), define the $L^{2}$ projection
$\Pi_{\mathcal{M}}^{k}v$ (resp. $\Pi_{\Sigma}^{k}v$) of $v$ onto
$P_{k}(\mathcal{M})$ (resp. $P_{k}(\Sigma)$) by
$(\Pi_{\mathcal{M}}^{k}v)|_{K}=\Pi_{K}^{k}v|_{K}$ in any cell
$K\in\mathcal{M}$ (resp. $(\Pi_{\Sigma}^{k}v)|_{S}=\Pi_{S}^{k}v$ along any
side $S\in\Sigma$). The mesh $\mathcal{M}$ gives rise to the piecewise
constant function $h_{\mathcal{M}}\in P_{0}(\mathcal{M})$ with
$h_{\mathcal{M}}|_{K}=h_{K}$ for any $K\in\mathcal{M}$;
$h_{\max}\coloneqq\max_{K\in\mathcal{M}}h_{K}$ is the maximal mesh-size of
$\mathcal{M}$.

###  3.2. HHO eigensolver

This subsection presents the HHO eigensolver for the computation of LEB. Let
$V_{\mathrm{nc}}\coloneqq H^{1}(\mathcal{M})$,
$a_{\mathrm{pw}}(u_{\mathrm{nc}},v_{\mathrm{nc}})\coloneqq(\nabla_{\mathrm{pw}}u_{\mathrm{nc}},\nabla_{\mathrm{pw}}v_{\mathrm{nc}})_{L^{2}(\Omega)}$,
and $W_{\mathrm{nc}}\coloneqq P_{k+1}(\mathcal{M})$.

####  3.2.1. Discrete ansatz spaces

Given a fixed $k\geq 0$, let $V_{h}\coloneqq P_{k+1}(\mathcal{M})\times
P_{k}(\Sigma(\Omega))$ denote the discrete ansatz space for $V$. In this
definition, $P_{k}(\Sigma(\Omega))$ is the subspace of $P_{k}(\Sigma)$ with
the convention $v_{\Sigma}\in P_{k}(\Sigma(\Omega))$ if $v_{\Sigma}|_{S}\equiv
0$ on boundary sides $S\in\Sigma(\partial\Omega)$ to model homogenous
Dirichlet boundary condition. For any $v_{h}=(v_{\mathcal{M}},v_{\Sigma})\in
V_{h}$, we use the notation $v_{K}\coloneqq v_{\mathcal{M}}|_{K}$ and
$v_{S}\coloneqq v_{\Sigma}|_{S}$ to abbreviate the restriction of
$v_{\mathcal{M}}$ in a cell $K\in\mathcal{M}$ and $v_{\Sigma}$ along a side
$S\in\Sigma$. The interpolation operator $\mathrm{I}_{h}:V\to V_{h}$ maps
$v\in V$ to
$\mathrm{I}_{h}v\coloneqq(\Pi_{\mathcal{M}}^{k+1}v,\Pi_{\Sigma}^{k}v)\in
V_{h}$.

####  3.2.2. Reconstruction operators and stabilization

We recall the potential reconstruction from [16].

##### _Potential reconstruction._

Given $v_{h}=(v_{\mathcal{M}},v_{\Sigma})\in V_{h}$, the potential
reconstruction $\mathcal{R}_{h}v_{h}\in W_{\mathrm{nc}}$ satisfies, for any
$\varphi_{k+1}\in W_{\mathrm{nc}}$, that

(3.4)
$\displaystyle\begin{split}&a_{\mathrm{pw}}(\mathcal{R}_{h}v_{h},\varphi_{k+1})=-(v_{\mathcal{M}},\Delta_{\mathrm{pw}}\varphi_{k+1})_{L^{2}(\Omega)}+\sum_{S\in\Sigma}(v_{S},[\nabla_{\mathrm{pw}}\varphi_{k+1}]_{S})_{L^{2}(S)}.\end{split}$

This defines $\mathcal{R}_{h}v_{h}$ uniquely up to piecewise constants, which
is fixed by

(3.5)
$\displaystyle\int_{K}\mathcal{R}_{h}v_{h}\,\mathrm{d}x=\int_{K}v_{K}\,\mathrm{d}x\quad\text{for
any }K\in\mathcal{M}.$

##### _Stabilization._

Given $K\in\mathcal{M}$ and a positive parameter $\sigma>0$, the local
stabilization $\mathrm{s}_{K}(u_{h},v_{h})$ reads, for any
$u_{h}=(u_{\mathcal{M}},u_{\Sigma}),v_{h}=(v_{\mathcal{M}},v_{\Sigma})\in
V_{h}$,

$\displaystyle\mathrm{s}_{K}(u_{h},v_{h})$ $\displaystyle\coloneqq\sigma
h_{K}^{-2}(u_{K}-\mathcal{R}_{h}u_{h},v_{K}-\mathcal{R}_{h}v_{h})_{L^{2}(K)}$
(3.6)
$\displaystyle\quad+\sigma\sum_{S\in\Sigma(K)}\ell(S,K)^{-1}(\Pi_{S}^{k}(u_{S}-\mathcal{R}_{h}u_{h}|_{K}),v_{S}-\mathcal{R}_{h}v_{h}|_{K})_{L^{2}(S)}$

with $\ell(S,K)$ from (3.2) and the global version
$\mathrm{s}_{h}(u_{h},v_{h})\coloneqq\sum_{K\in\mathcal{M}}\mathrm{s}_{K}(u_{h},v_{h})$.
A similar stabilization has been proposed (and analysed) in [13, Example 2.8]
for the discrete ansatz space $P_{k}(\mathcal{M})\times
P_{k}(\Sigma(\Omega))$, however we note that it is not sufficient for
stability for the present ansatz space.

##### _Discrete eigenvalue problem._

The discrete eigenvalue problem (1.5) is defined with

(3.7) $\displaystyle a_{h}(u_{h},v_{h})\coloneqq
a_{\mathrm{pw}}(\mathcal{R}_{h}u_{h},\mathcal{R}_{h}v_{h})\quad\text{and}\quad
b_{h}(u_{h},v_{h})=(u_{\mathcal{M}},v_{\mathcal{M}})_{L^{2}(\Omega)}$

for any
$u_{h}=(u_{\mathcal{M}},u_{\Sigma}),v_{h}=(v_{\mathcal{M}},v_{\Sigma})\in
V_{h}$. It is straightforward to verify the following result.

######  Lemma 3.2 (coercivity).

The bilinear form $a_{h}+\mathrm{s}_{h}$ is a scalar product in $V_{h}$. In
particular, there are $\mathrm{dim}(P_{k+1}(\mathcal{M}))$ finite discrete
eigenvalues of (1.5).∎

###  3.3. Lower eigenvalue bounds

This subsection verifies (A)–(C) as outlined in Subsection 1.3 and derives LEB
for the Laplace operator. The Galerkin projection is defined by (1.8)–(1.9).

######  Corollary 3.3 (LEB for Laplace).

It holds

$\mathrm{LEB}(j)\coloneqq\min\\{1,1/(\sigma/\pi^{2}+\sigma
c_{\mathrm{tr}}+h_{\mathrm{max}}^{2}\lambda_{h}(j)/\pi^{2})\\}\lambda_{h}(j)\leq\lambda(j).$

###### Proof.

Given $v\in V$, the property $\mathrm{G}_{h}v=\mathcal{R}_{h}\mathrm{I}_{h}v$
in (1.7) follows from the orthogonality
$v-\mathcal{R}_{h}\mathrm{I}_{h}v\perp_{a_{\mathrm{pw}}}W_{\mathrm{nc}}$ [16,
Eq. (18)] and
$\Pi_{\mathcal{M}}^{0}\mathcal{R}_{h}\mathrm{I}_{h}v=\Pi_{\mathcal{M}}^{0}v$
in (3.5). Hence, the Pythagoras theorem implies

$\displaystyle\|\mathrm{I}_{h}v\|_{a}^{2}=\|v\|^{2}_{a}-\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}.$

This is (A). Fix a cell $K\in\mathcal{M}$. The best approximation property of
$L^{2}$ projections and (1.7) provide the upper bound

$\displaystyle\sigma
h_{K}^{-2}\|\Pi_{K}^{k+1}(v-\mathrm{G}_{h}v)\|_{L^{2}(K)}^{2}+\sigma\sum_{S\in\Sigma(K)}\ell(S,K)^{-1}\|\Pi_{S}^{k}(v-\mathrm{G}_{h}v|_{K})\|_{L^{2}(S)}^{2}$
$\displaystyle\leq\sigma
h_{K}^{-2}\|v-\mathrm{G}_{h}v\|_{L^{2}(K)}^{2}+\sigma\sum_{S\in\Sigma(K)}\ell(S,K)^{-1}\|v-\mathrm{G}_{h}v|_{K}\|_{L^{2}(S)}^{2}$

for the stabilization $\mathrm{s}_{K}(\mathrm{I}_{h}v,\mathrm{I}_{h}v)$. This,
the Poincaré, and trace inequality from Lemma 3.1 establish

(3.8)
$\displaystyle\|\mathrm{I}_{h}v\|_{\mathrm{s}_{h}}^{2}\leq(\sigma/\pi^{2}+\sigma
c_{\mathrm{tr}})\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2},$

which is (B) with $\alpha\coloneqq(\sigma/\pi^{2}+\sigma c_{\mathrm{tr}})$.
The Pythagoras theorem and the best-approximation property of $L^{2}$
projections prove

(3.9)
$\displaystyle\|\mathrm{I}_{h}v\|^{2}_{b}=\|\Pi_{\mathcal{M}}^{k+1}v\|^{2}=\|v\|^{2}-\|(1-\Pi_{\mathcal{M}}^{k+1})v\|^{2}\geq\|v\|^{2}-\|(1-\mathrm{G}_{h})v\|^{2}.$

This and the Poincaré inequality show (C) with $\beta\coloneqq
h_{\max}^{2}/\pi^{2}$. Theorem 2.1 concludes the proof with the constants
$\alpha$, $\beta$ from above. ∎

###  3.4. Numerical example

The computer experiments throughout this paper are carried out on regular
triangulation into simplices. Some general remarks on the realization precede
the numerical results.

####  3.4.1. Displayed quantities

The convergence history plots display the error
$\lambda_{\mathrm{C}}(j)-\mathrm{LEB}(j)$ against the number of degrees of
freedom $\mathrm{ndof}$ in a log-log plot, where the upper bound
$\lambda_{\mathrm{C}}(j)$ is the Rayleigh quotient of some conforming function
$u_{\mathrm{C}}(j)\in P_{k+1}(\mathcal{M})\cap V$. If $j=1$, then
$u_{\mathrm{C}}(j)$ is the scaled nodal average of $\mathcal{R}_{h}u_{h}(j)$
with $\|u_{\mathrm{C}}\|_{b}=1$ and for $j\geq 1$, $u_{\mathrm{C}}(j)$ can be
obtained from the conforming Galerkin $P_{k+1}$ FEM on the same mesh. At least
on simplicial meshes (without hanging nodes),
$\|\mathrm{I}_{h}v_{\mathrm{C}}\|_{a_{h}}^{2}+\|\mathrm{I}_{h}v_{\mathrm{C}}\|_{\mathrm{s}_{h}}^{2}=\|v_{\mathrm{C}}\|_{a}^{2}$
and $\|\mathrm{I}_{h}v_{\mathrm{C}}\|_{b_{h}}^{2}=\|v_{\mathrm{C}}\|_{b}^{2}$
holds for any $v_{\mathrm{C}}\in P_{k+1}(\mathcal{M})\cap V$. Therefore, the
discrete Rayleigh-Ritz principle (1.6) shows
$\lambda_{h}(j)\leq\lambda_{\mathrm{C}}(j)$. In combination with Theorem 2.1,
$\mathrm{LEB}(j)\leq\\{\lambda,\lambda_{h}(j)\\}\leq\lambda_{\mathrm{C}}(j)$
and so,
$|\lambda(j)-\lambda_{h}(j)|\leq\lambda_{\mathrm{C}}(j)-\mathrm{LEB}(j)$.

####  3.4.2. Adaptive refinement indicator

If $\mathrm{LEB}(j)=\lambda_{h}(j)$, the error estimator
$\lambda_{\mathrm{C}}(j)-\lambda_{h}(j)$ can be localized following [23, Lemma
6.3]. Let $u_{\mathrm{C}}(j)$ be given such
$\lambda_{\mathrm{C}}(j)=\|u_{\mathrm{C}}\|_{a}^{2}$ with the normalization
$\|u_{\mathrm{C}}(j)\|_{b}^{2}=1$ and
$(u_{\mathrm{C}}(j),u_{\mathcal{M}}(j))_{L^{2}(\Omega)}>0$, where
$u_{h}(j)=(u_{\mathcal{M}}(j),u_{\Sigma}(j))\in V_{h}$ is the computed
eigenvector of $\lambda_{h}(j)$. Elementary algebra with
$\|\mathcal{R}_{h}u_{h}(j)\|^{2}_{a_{\mathrm{pw}}}=\lambda_{h}(j)-\|u_{h}(j)\|_{\mathrm{s}_{h}}^{2}$
shows

$\displaystyle\lambda_{\mathrm{C}}(j)-\lambda_{h}(j)=\|u_{\mathrm{C}}(j)\|_{a}^{2}-\lambda_{h}(j)$
$\displaystyle\quad=\|u_{\mathrm{C}}(j)-\mathcal{R}_{h}u_{h}(j)\|^{2}_{a_{\mathrm{pw}}}+2a_{\mathrm{pw}}(u_{\mathrm{C}}(j),\mathcal{R}_{h}u_{h}(j))-\|\mathcal{R}_{h}u_{h}(j)\|^{2}_{a_{\mathrm{pw}}}-\lambda_{h}(j)$
$\displaystyle\quad=\|u_{\mathrm{C}}(j)-\mathcal{R}_{h}u_{h}(j)\|^{2}_{a_{\mathrm{pw}}}+2a_{\mathrm{pw}}(u_{\mathrm{C}}(j),\mathcal{R}_{h}u_{h}(j))-2\lambda_{h}(j)+\|u_{h}(j)\|_{\mathrm{s}_{h}}^{2}.$

The discrete variational formulation (1.5) and the commuting property (1.7)
show

$\displaystyle 2a_{\mathrm{pw}}(u_{\mathrm{C}}(j),\mathcal{R}_{h}u_{h}(j))$
$\displaystyle=2\lambda_{h}(u_{\mathcal{M}},u_{\mathrm{C}}(j))_{L^{2}(\Omega)}$
$\displaystyle=-\lambda_{h}\|u_{\mathrm{C}}(j)-u_{\mathcal{M}}(j)\|^{2}+\lambda_{h}(\|u_{\mathrm{C}}\|^{2}_{b}+\|u_{h}(j)\|_{b_{h}}^{2}).$

The combination of the two previously displayed formula with
$\|u_{\mathrm{C}}\|^{2}_{b}+\|u_{h}(j)\|_{b_{h}}^{2}=2$ concludes

(3.10)
$\displaystyle\lambda_{\mathrm{C}}(j)-\lambda_{h}(j)\leq\eta\coloneqq\|u_{\mathrm{C}}(j)-\mathcal{R}_{h}u_{h}(j)\|^{2}_{a_{\mathrm{pw}}}+\|u_{h}(j)\|^{2}_{\mathrm{s}_{h}}.$

Notice that the term
$-\lambda_{h}\|u_{\mathrm{C}}(j)-u_{\mathcal{M}}(j)\|^{2}\leq 0$ is neglected
in the last formula. However, it is expected that
$\|u_{\mathrm{C}}(j)-u_{\mathcal{M}}(j)\|^{2}$ is of higher order due to
additional smoothness of the exact eigenvector. A generalization of (3.10) to
general polyhedral meshes is possible with the addition of
$\|\mathrm{I}_{h}u_{\mathrm{C}}\|_{\mathrm{s}_{h}}^{2}$ on the right-hand side
of (3.10). The bound (3.10) motivates the local refinement indicator

$\displaystyle\eta(K)\coloneqq\|\nabla_{\mathrm{pw}}(u_{\mathrm{C}}(j)-\mathcal{R}_{h}u_{h}(j))\|^{2}_{L^{2}(K)}+\mathrm{s}_{K}(u_{h}(j),u_{h}(j)).$

This localization is relevant for simple eigenvalues. For the approximation of
eigenvalue clusters, we suggest the use of residual-based error estimators,
e.g., from [7], as in [19].

####  3.4.3. Adaptive algorithm

Whenever $\alpha+\beta\lambda_{h}(j)>1$, the mesh is refined uniformly.
Otherwise, the refinement indicator $\eta(K)$ is utilized in the standard
Dörfler marking with bulk parameter $1/2$, i.e., at each refinement step, a
subset $\mathfrak{M}\subset\mathcal{M}$ of minimal cardinality is selected so
that

$\displaystyle\eta\leq\frac{1}{2}\sum\nolimits_{K\in\mathfrak{M}}\eta(K).$

Figure 1. (a) convergence history plot of
$\lambda_{\mathrm{C}}(1)-\mathrm{LEB}(1)$ and (b) adaptive mesh with 736
triangles ($k=2$) for the Laplace eigenvalue problem in Subsection 3.4.4

####  3.4.4. Computer benchmark

This benchmark approximates the first Laplace eigenvalue in the $L$-shaped
domain $\Omega\coloneqq(-1,1)\setminus([0,1]\times[-1,0])$ with the reference
value $\lambda(1)=9.63972384$ from the bound
$\mathrm{LEB}(1)\leq\lambda(1)\leq\lambda_{\mathrm{C}}(1)$. We chose
$\sigma\coloneqq 2^{-1}(\pi^{-2}+c_{\mathrm{tr}})^{-1}=0.9598$ in Corollary
3.3, which provides
$\mathrm{LEB}(j)=\min\\{1,1/(1/2+h_{\mathrm{max}}^{2}\lambda_{h}(j)/\pi^{2})\\}\lambda_{h}(j)\leq\lambda(j)$.
The observations in this example also applies to all other benchmarks in
Section 4–5 below. Figure 1(a) displays the convergence history of
$\lambda_{\mathrm{C}}(1)-\mathrm{LEB}(1)$. Uniform mesh refinements leads to
the suboptimal convergence rates $2/3$ due to the expected singularity of the
first eigenvector. Adaptive mesh computation refines towards the origin as
shown in Figure 1(b) and recovers the optimal convergence rates $k+1$ for all
displayed polynomial degrees $k$. Undisplayed numerical experiments show that
there are only marginal differences between $\eta$ and
$\lambda_{\mathrm{C}}(1)-\mathrm{LEB}(1)$ except on very coarse meshes. Notice
that Figure 1(a) also clearly displays a disadvantage of this method in
comparison to other numerical schemes [22, 6, 20] with postprocessed
eigenvalue bounds. The constants in local compact embeddings (e.g., trace
inequalities) provide an upper bound for $\sigma$ so that $\alpha<1$. Since
overestimation of these constants is expected, $\sigma$ may be rather small in
comparison to the best possible value with direct impact on the numerical
method. We observe a preasymptotic range, where the discrete eigenvalue and
lower bound are close to zero (and thus, not relevant) until the stabilization
is resolved. A remedy is the computation of accurate upper bounds of the
involved constants; we refer to Section 6 for further details.

##  4\. Steklov eigenvalue problem

The Steklov eigenvalue problem seeks eigenpairs $(\lambda,u)$ with

$\displaystyle-\Delta u+u=0\text{ in }\Omega\quad\text{and}\quad\partial
u/\partial\nu=\lambda u\text{ on }\partial\Omega.$

The weak formulation of this is (1.2) with $V\coloneqq H^{1}(\Omega)$,
$a(u,v)\coloneqq(\nabla u,\nabla v)_{L^{2}(\Omega)}+(u,v)_{L^{2}(\Omega)}$,
and $b(u,v)\coloneqq(u,v)_{L^{2}(\partial\Omega)}$.

###  4.1. HHO eigensolver

Let $V_{\mathrm{nc}}\coloneqq H^{1}(\mathcal{M})$,
$a_{\mathrm{pw}}\coloneqq(\nabla_{\mathrm{pw}}\bullet,\nabla_{\mathrm{pw}}\bullet)_{L^{2}(\Omega)}$,
and $W_{\mathrm{nc}}(\mathcal{M})\coloneqq P_{k+1}(\mathcal{M})$ with $k\geq
0$. We utilize the discrete ansatz space $V_{h}\coloneqq
P_{k}(\mathcal{M})\times P_{k+1}(\Sigma)$ with the interpolation
$\mathrm{I}_{h}v\coloneqq(\Pi_{\mathcal{M}}^{k}v,\Pi_{\Sigma}^{k+1}v)\in
V_{h}$ for any $v\in V$. The discrete problem (1.5) is defined with the
bilinear forms

$\displaystyle a_{h}(u_{h},v_{h})$
$\displaystyle\coloneqq(\nabla_{\mathrm{pw}}\mathcal{R}_{h}u_{h},\nabla_{\mathrm{pw}}\mathcal{R}_{h}v_{h})_{L^{2}(\Omega)}+(u_{\mathcal{M}},v_{\mathcal{M}})_{L^{2}(\Omega)},$
$\displaystyle b_{h}(u_{h},v_{h})$
$\displaystyle\coloneqq(u_{\Sigma},v_{\Sigma})_{L^{2}(\partial\Omega)},$

and the stabilization
$\mathrm{s}_{h}(u_{h},v_{h})\coloneqq\sum_{K\in\mathcal{M}}\mathrm{s}_{K}(u_{h},v_{h})$,

$\displaystyle\mathrm{s}_{K}(u_{h},v_{h})$ $\displaystyle\coloneqq\sigma
h_{K}^{-2}(\Pi_{K}^{k}(u_{K}-\mathcal{R}_{h}u_{h}),v_{K}-\mathcal{R}_{h}v_{h})_{L^{2}(\Omega)}$
$\displaystyle\quad+\sigma\sum_{S\in\Sigma(K)}\ell(S,K)^{-1}(u_{S}-\mathcal{R}_{h}u_{h}|_{K},v_{S}-\mathcal{R}_{h}v_{h}|_{K})_{L^{2}(S)},$

for any
$u_{h}=(u_{\mathcal{M}},u_{\Sigma}),v_{h}=(v_{\mathcal{M}},v_{\Sigma})\in
V_{h}$, where the potential reconstruction $\mathcal{R}_{h}$ is defined
verbatim as in (3.4)–(3.5). It is straightforward to verify the coercivity of
$a_{h}+\mathrm{s}_{h}$.

######  Lemma 4.1 (coercivity).

The bilinear form $a_{h}+\mathrm{s}_{h}$ is a scalar product in $V_{h}$. In
particular, there are $\mathrm{dim}(P_{k+1}(\Sigma(\partial\Omega)))$ finite
discrete eigenvalues of (1.5).∎

###  4.2. Lower eigenvalue bounds

This subsection verifies (A)–(C) and derives LEB for the Steklov eigenvalue
problem. The Galerkin projection is defined by (1.8)–(1.9). We utilize the
following trace inequality. Given a boundary side
$S\in\Sigma(\partial\Omega)$, there exists a unique cell $K\in\mathcal{M}$
with $S\in\Sigma(K)$. The convex hull of $S$ and a point $x\in K$ of largest
volume is denoted by $K^{\mathrm{m}}_{S}$. (Notice that $K_{S}^{\mathrm{m}}=K$
for simplicial meshes.) The trace inequality (3.3) in $K_{S}^{\mathrm{m}}$ and
a Hölder inequality provide, for any $v\in H^{1}(K)$, that

(4.1)
$\displaystyle\|v\|_{L^{2}(S)}^{2}\leq\frac{|S|}{|K_{S}^{\mathrm{m}}|}\|v\|_{L^{2}(K)}^{2}+\frac{2h_{K_{S}^{\mathrm{m}}}|S|}{n|K_{S}^{\mathrm{m}}|}\|v\|_{L^{2}(K)}\|\nabla
v\|_{L^{2}(K)}.$

This motivates the definition of the constant

(4.2) $\displaystyle\beta_{\mathrm{St}}\coloneqq
c_{\mathrm{0}}\max_{S\in\Sigma(\partial\Omega)}\frac{h_{K}|S|}{|K_{S}^{\mathrm{m}}|}(h_{K}/\pi^{2}+2h_{K_{S}^{\mathrm{m}}}/(n\pi))$

utilized in Corollary 4.2 below, where $c_{0}$ is the maximal number of
boundary sides belonging to the same cell.

######  Corollary 4.2 (LEB for Steklov).

It holds

(4.3) $\displaystyle\mathrm{LEB}(j)\coloneqq\min\\{1,1/(\sigma/\pi^{2}+\sigma
c_{\mathrm{tr}}+\beta_{\mathrm{St}}\lambda_{h}(j))\\}\leq\lambda(j).$

###### Proof.

The property (1.7) is independent of boundary data and holds here as well.
This, the Pythagoras theorem with the orthogonality
$v-\mathrm{G}_{h}v\perp_{a_{\mathrm{pw}}}W_{\mathrm{nc}}$, and the best
approximation property of $\Pi_{\mathcal{M}}^{k}$ imply, for any $v\in V$,
that

$\displaystyle\|\mathrm{I}_{h}v\|_{a_{h}}^{2}$
$\displaystyle=\|\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}+\|\Pi_{\mathcal{M}}^{k}v\|^{2}$
$\displaystyle=\|v\|_{a_{\mathrm{pw}}}^{2}-\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}+\|\Pi_{\mathcal{M}}^{k}v\|^{2}\leq\|v\|_{a}^{2}-\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2},$

which is (A). The same arguments as in the proof of Corollary 3.3 provide (B)
with $\alpha\coloneqq\sigma/\pi^{2}+\sigma c_{\mathrm{tr}}$. For any boundary
side $S\in\partial\Omega$, let $K$ denote the unique cell with
$S\in\Sigma(K)$. The Pythagoras theorem and the best approximation property of
$L^{2}$ projections imply

$\displaystyle\|\Pi_{S}^{k+1}v\|^{2}_{L^{2}(S)}=\|v\|_{L^{2}(S)}^{2}-\|v-\Pi_{S}^{k+1}v\|_{L^{2}(S)}^{2}\leq\|v\|_{L^{2}(S)}^{2}-\|v-\mathrm{G}_{h}v\|_{L^{2}(S)}^{2}.$

Invoking the trace inequality (4.1) and the Poincaré inequality, we infer that

(4.4)
$\displaystyle\|v-\mathrm{G}_{h}v\|_{L^{2}(S)}^{2}\leq\frac{h_{K}|S|}{|K_{S}^{\mathrm{m}}|}(h_{K}/\pi^{2}+2h_{K_{S}^{\mathrm{m}}}/(n\pi))\|\nabla(v-\mathrm{G}_{h}v)\|_{L^{2}(K)}^{2}.$

The combination of the two previously displayed formula with an overlap
argument results in

$\displaystyle\|\mathrm{I}_{h}v\|_{b_{h}}^{2}=\|\Pi_{\Sigma}^{k+1}v\|_{L^{2}(\partial\Omega)}^{2}\leq\|v\|_{b}^{2}-\beta_{\mathrm{St}}\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2},$

whence (C) holds with $\beta\coloneqq\beta_{\mathrm{St}}$. Theorem 2.1
concludes the assertion. ∎

Figure 2. Convergence history plot of
$\lambda_{\mathrm{C}}(1)-\mathrm{LEB}(1)$ for the Steklov eigenvalue problem
in Subsection 4.3

###  4.3. Computer benchmark

This benchmark approximates the first Steklov eigenvalue in the $L$-shaped
domain $\Omega\coloneqq(-1,1)\setminus([0,1]\times[-1,0])$ with the reference
value $\lambda(1)=0.34141604251$ from the bound
$\mathrm{LEB}(1)\leq\lambda(1)\leq\lambda_{\mathrm{C}}(1)$. We set
$\sigma\coloneqq 2^{-1}(\pi^{-2}+c_{\mathrm{tr}})^{-1}=0.9598$ so that
$\mathrm{LEB}(1)=\min\\{1,1/(1/2+\beta_{\mathrm{st}}\lambda_{h}(1))\\}\lambda_{h}(1)\leq\lambda(1)$
from Corollary 4.2. The computation of $\lambda_{\mathrm{C}}(1)$ and
$u_{\mathrm{C}}(1)$ follows Subsection 3.4.4. For the first discrete
eigenvalue $u_{h}(1)=(u_{\mathcal{M}}(1),u_{\Sigma}(1))$, the bound

$\displaystyle\lambda_{\mathrm{C}}(1)-\lambda_{h}(1)\leq\|u_{\mathrm{C}}(1)-\mathcal{R}_{h}u_{h}(1)\|^{2}_{a_{\mathrm{pw}}}+\|u_{\mathrm{C}}(1)-u_{\mathcal{M}}(1)\|^{2}+\|u_{h}(1)\|_{\mathrm{s}_{h}}^{2}$

similar to (3.10) provides a refinement indicator by localization of the
right-hand side. Figure 2 displays the convergence history of
$\lambda_{\mathrm{C}}(1)-\lambda_{h}(1)$ and the observations in Subsection
3.4.4 apply. An advantage of this method over [24, 20] is that, even for
smooth eigenfunctions, the lower bounds therein can only converge towards the
exact eigenvalue with the rate $1/2$ due to the scaling of trace inequalities.

##  5\. Linear elasticity

Let the boundary $\partial\Omega$ of $\Omega$ be split into a closed connected
nonempty Dirichlet part $\Gamma_{\mathrm{D}}\subset\partial\Omega$ and a
Neumann part
$\Gamma_{\mathrm{D}}\coloneqq\partial\Omega\setminus\Gamma_{\mathrm{D}}$. The
linear elasticity eigenvalue problem seeks eigenpairs $(\lambda,u)$ such that

(5.1) $\displaystyle-\mathrm{div}\,\sigma=\lambda u\text{ in
}\Omega,\quad\sigma=\mathbb{C}\varepsilon(u),\quad u=0\text{ on
}\Gamma_{\mathrm{D}},\quad\text{and }\sigma\nu=0\text{ on
}\Gamma_{\mathrm{N}}.$

Here, $\varepsilon(v)\coloneqq\mathrm{sym}(\mathrm{D}v)$ is the symmetric part
of the gradient $\mathrm{D}v$ of $v$ and $\mathbb{C}A\coloneqq 2\mu
A+\kappa\mathrm{tr}(A)\mathrm{I}_{n}$ for any $A\in\mathbb{S}$ denotes the
linearized strain tensor with Lamé parameters $\lambda,\mu>0$. The weak
formulation of (5.1) is (1.2) with $V\coloneqq\\{v\in
H^{1}(\Omega)^{n}:v=0\text{ on }\Gamma_{\mathrm{D}}\\}$,
$a(u,v)\coloneqq(\mathbb{C}\varepsilon(u),\varepsilon(v))_{L^{2}(\Omega)}$,
and $b(u,v)\coloneqq(u,v)_{L^{2}(\Omega)}$.

###  5.1. HHO eigensolver

Assume that the Dirichlet boundary can be exactly resolved by the mesh
$\mathcal{M}$. Let $V_{\mathrm{nc}}\coloneqq H^{1}(\mathcal{M})^{n}$,
$a_{\mathrm{pw}}\coloneqq(\mathbb{C}\varepsilon_{\mathrm{pw}}(\bullet),\varepsilon_{\mathrm{pw}}(\bullet))_{L^{2}(\Omega)}$,
and $W_{\mathrm{nc}}\coloneqq P_{k+1}(\mathcal{M})^{n}$. Given any $k\geq 1$,
$V_{h}\coloneqq P_{k+1}(\mathcal{M})^{n}\times P_{k}(\mathcal{M})^{n}$ denotes
the discrete ansatz space. The potential reconstruction
$\mathcal{R}_{h}v_{h}\in W_{\mathrm{nc}}$ of
$v_{h}=(v_{\mathcal{M}},v_{\Sigma})\in V_{h}$ satisfies, for any
$\varphi_{\mathrm{nc}}\in W_{\mathrm{nc}}$, that

$\displaystyle a_{\mathrm{pw}}(\mathcal{R}_{h}v_{h},\varphi_{\mathrm{nc}})$
$\displaystyle=-(v_{\mathcal{M}},\mathrm{div}_{\mathrm{pw}}\mathbb{C}\varepsilon_{\mathrm{pw}}(\varphi_{\mathrm{nc}}))_{L^{2}(\Omega)}$
(5.2)
$\displaystyle\qquad+\sum_{S\in\Sigma}(v_{S},[\mathbb{C}\varepsilon_{\mathrm{pw}}(\varphi_{\mathrm{nc}})]_{S}\nu_{S})_{L^{2}(S)}.$

This defines $\mathcal{R}_{h}u_{h}$ uniquely up to the degrees of freedom
associated with rigid body motions, which are fixed, for any
$K\in\mathcal{M}$, by

(5.3)
$\displaystyle\int_{K}\mathcal{R}_{h}v_{h}\,\mathrm{d}x=\int_{K}v_{K}\,\mathrm{d}x,\int_{K}\mathrm{D}_{\mathrm{ss}}\mathcal{R}_{h}v_{h}\,\mathrm{d}x=\sum_{S\in\Sigma(K)}\int_{S}\mathrm{asym}(\nu_{K}\otimes
v_{S})\,\mathrm{d}s,$

where $\mathrm{D}_{\mathrm{ss}}$ denotes the asymmetric part of the gradient
and $\nu_{K}\otimes v_{S}\coloneqq\nu_{K}v_{S}^{t}\in\mathbb{R}^{n\times n}$.
By design, $\mathcal{R}_{h}$ satisfies the following.

######  Lemma 5.1 (projection property).

Any $v\in V$ satisfies the $a_{\mathrm{pw}}$ orthogonality
$v-\mathcal{R}_{h}\mathrm{I}_{h}v\perp W_{\mathrm{nc}}$.

###### Proof.

For any $\varphi_{nc}\in W_{\mathrm{nc}}$,
$\mathrm{div}_{\mathrm{pw}}\mathbb{C}\varepsilon_{\mathrm{pw}}(\varphi_{nc})\in
P_{k-1}(\mathcal{M})^{n}$ and
$[\mathbb{C}\varepsilon_{\mathrm{pw}}(\varphi_{nc})]_{S}\nu_{S}\in
P_{k}(S)^{n}$ along $S\in\Sigma$. By choosing $v_{h}\coloneqq\mathrm{I}_{h}v$
in (5.1), we observe that $v_{\mathcal{M}}=\Pi_{\mathcal{M}}^{k+1}v$ and
$v_{S}=\Pi_{S}^{k}v$ on the right-hand side of (5.1) can be replaced by $v$. A
piecewise integration by parts of the resulting term shows
$(\mathbb{C}\varepsilon_{\mathrm{pw}}(\mathcal{R}_{h}\mathrm{I}_{h}v),\varepsilon_{\mathrm{pw}}(\varphi_{nc}))_{L^{2}(\Omega)}=(\mathbb{C}\varepsilon(v),\varepsilon_{\mathrm{pw}}(\varphi_{nc}))_{L^{2}(\Omega)}$.
∎

The discrete problem (1.5) is defined with

$\displaystyle a_{h}(u_{h},v_{h})\coloneqq
a_{\mathrm{pw}}(\mathcal{R}_{h}u_{h},\mathcal{R}_{h}v_{h}),\quad
b_{h}(u_{h},v_{h})\coloneqq(u_{\mathcal{M}},v_{\mathcal{M}})_{L^{2}(\Omega)},$

and
$\mathrm{s}_{h}(u_{h},v_{h})\coloneqq\sum_{K\in\mathcal{M}}\mathrm{s}_{K}(u_{h},v_{h})$
with

$\displaystyle\mathrm{s}_{K}(u_{h},v_{h})\coloneqq
2\mu\sigma\int_{K}h_{K}^{-2}(u_{K}-\mathcal{R}_{h}u_{h})\cdot(v_{K}-\mathcal{R}_{h}v_{h})\,\mathrm{d}x$
$\displaystyle\quad+2\mu\sigma\sum_{S\in\Sigma(K)}\ell(S,K)^{-1}\int_{S}\Pi_{S}^{k}(u_{S}-\mathcal{R}_{h}u_{h}|_{K})\cdot\Pi_{S}^{k}(v_{S}-\mathcal{R}_{h}v_{h}|_{K})\,\mathrm{d}s$

for any
$u_{h}=(u_{\mathcal{M}},u_{\Sigma}),v_{h}=(v_{\mathcal{M}},v_{\Sigma})\in
V_{h}$, where $\ell(S,K)$ is the weight from (3.2). It is straightforward to
verify the coercivity of $a_{h}+\mathrm{s}_{h}$.

######  Lemma 5.2 (coercivity).

The bilinear form $a_{h}+\mathrm{s}_{h}$ is a scalar product in $V_{h}$. In
particular, there are $\mathrm{dim}(P_{k+1}(\mathcal{M}))$ finite discrete
eigenvalues of (1.5). ∎

###  5.2. Lower eigenvalue bounds

This subsection verifies (A)–(C) and derives LEB for the linear elasticity
eigenvalue problem. The Galerkin projection $\mathrm{G}_{h}v\in
W_{\mathrm{nc}}$ of $v\in V_{\mathrm{nc}}$ is the unique solution to (1.8)
with, for any $K\in\mathcal{M}$,

(5.4)
$\displaystyle\int_{K}\mathrm{G}_{h}v\,\mathrm{d}x=\int_{K}v\,\mathrm{d}x\quad\text{and}\quad\int_{K}\mathrm{D}_{\mathrm{ss}}\mathrm{G}_{h}v_{h}\,\mathrm{d}x=\int_{K}\mathrm{D}_{\mathrm{ss}}v\,\mathrm{d}s.$

######  Lemma 5.3 ($\mathcal{R}_{h}\circ\mathrm{I}_{h}=\mathrm{G}_{h}$).

The identity (1.7) holds in $V$.

###### Proof.

Given $v\in V$, Lemma 5.1 implies that $\mathcal{R}_{h}\mathrm{I}_{h}v$
satisfies (1.8). For any $K\in\mathcal{M}$,
$\int_{K}\mathcal{R}_{h}\mathrm{I}_{h}v\,\mathrm{d}x=\int_{K}\Pi_{\mathcal{M}}^{k+1}v\,\mathrm{d}x=\int_{K}v\,\mathrm{d}x$
and
$\sum_{S\in\Sigma(K)}\int_{S}\mathrm{asym}(\nu_{K}\otimes\Pi_{S}^{k}v)\,\mathrm{d}s=\sum_{S\in\Sigma(K)}\int_{S}\mathrm{asym}(\nu_{K}\otimes
v)\,\mathrm{d}s=\int_{K}\mathrm{D}_{\mathrm{ss}}v\,\mathrm{d}x$ from (5.3) and
an integration by parts. Hence,
$\mathcal{R}_{h}\mathrm{I}_{h}v=\mathrm{G}_{h}v$. ∎

Given $K\in\mathcal{M}$, let $c_{\mathrm{Korn}}(K)$ denote the Korn constant
in the inequality

(5.5) $\displaystyle\|\mathrm{D}v\|_{L^{2}(K)}\leq
c_{\mathrm{Korn}}(K)\|\varepsilon(v)\|_{L^{2}(K)}$

for any $v\in H^{1}(K)^{n}$ with
$\int_{K}\mathrm{D}_{\mathrm{ss}}v\,\mathrm{d}x=0$ [3]. The computation of LEB
for linear elasticity requires the following result in the local space
$V(K)\coloneqq\\{v\in H^{1}(K)^{n}:\int_{K}v\,\mathrm{d}x=0\text{ and
}\int_{K}\mathrm{D}_{\mathrm{ss}}v\,\mathrm{d}x=0\\}$.

######  Lemma 5.4 (local bound in linear elasticity).

Given a convex polyhedra $K\subset\mathbb{R}^{n}$, there exists a constant
$\gamma(K)\geq c_{\mathrm{Korn}}^{-2}(K)(\pi^{-2}+c_{\mathrm{st}})^{-1}$ with,
for any $v\in V(K)$,

$\displaystyle\sum_{S\in\Sigma(K)}\ell(S,K)^{-1}\|v\|_{L^{2}(S)}^{2}+h_{K}^{-2}\|v\|_{L^{2}(K)}^{2}\leq\gamma(K)^{-1}\|\varepsilon(v)\|^{2}_{L^{2}(K)}.$

The constant $\gamma(K)$ may depend on the shape but not on the diameter of
$K$.

###### Proof.

The assertion follows from Lemma 3.1, a Poincaré, and Korn inequality. ∎

Lemma 5.3–5.4 allow for the verification of (A)–(C) with the constants
$c_{\mathrm{Korn}}\coloneqq\max_{K\in\mathcal{M}}c_{\mathrm{Korn}}(K)$ and
$\gamma\coloneqq\min_{K\in\mathcal{M}}\gamma(K)$.

######  Corollary 5.5 (LEB for linear elasticity).

It holds

$\displaystyle\min\\{1,1/(\gamma^{-1}\sigma+c_{\mathrm{Korn}}^{2}h_{\mathrm{max}}^{2}\lambda_{h}(j)/(2\pi^{2}\mu))\\}\lambda_{h}(j)\leq\lambda(j)$

###### Proof.

Given $v\in V$, the Pythagoras theorem with the $a_{\mathrm{pw}}$
orthogonality $v-\mathrm{G}_{h}v\perp W_{\mathrm{nc}}$ from Lemma 5.1 and
Lemma 5.3 proves (A), namely,

$\displaystyle\|\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}=\|v\|_{a}^{2}-\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}.$

Since $v-\mathrm{G}_{h}v\in V(K)$ for any $K\in\mathcal{M}$, Lemma 5.3–5.4 and
the best approximation property of $L^{2}$ projections provide

$\displaystyle s_{K}(\mathrm{I}_{h}v,\mathrm{I}_{h}v)$ $\displaystyle\leq
2\mu\sigma\sum_{S\in\Sigma(K)}\ell(S,K)^{-1}\|v-\mathrm{G}_{h}v\|^{2}_{L^{2}(S)}+2\mu\sigma
h_{K}^{-2}\|v-\mathrm{G}_{h}v\|_{L^{2}(K)}^{2}$ (5.6) $\displaystyle\leq
2\gamma(K)^{-1}\mu\sigma\|\varepsilon(v-\mathrm{G}_{h}v)\|^{2}_{L^{2}(K)}.$

The sum of this over all $K\in\mathcal{M}$ yields (B) with the constant
$\alpha\coloneqq\gamma^{-1}\sigma$. The combination of (3.9) with the Korn
inequality (5.5) proves (C) with the constant
$\beta=c_{\mathrm{Korn}}^{2}h_{\mathrm{max}}^{2}/(2\pi^{2}\mu)$. Theorem 2.1
concludes the proof. ∎

######  Remark 5.6 ($\kappa$ robustness).

Utilizing the arguments of [10], we can prove the robustness of this method
with respect to $\kappa\to\infty$ on regular triangulations $\mathcal{M}$ into
simplices in the following sense. Given $f\in L^{2}(\Omega)^{n}$, let $u\in V$
solve

$\displaystyle-\mathrm{div}\,\sigma=f\text{ in
}\Omega,\quad\sigma=\mathbb{C}\varepsilon(u),\quad\sigma\nu=0\text{ on
}\Gamma_{\mathrm{N}}.$

Then the discrete solution $u_{h}\in V_{h}$ to
$a_{h}(u_{h},v_{h})+\mathrm{s}_{h}(u_{h},v_{h})=(f,v_{\mathcal{M}})_{L^{2}(\Omega)}$
for any $v_{h}=(v_{\mathcal{M}},v_{\Sigma})\in V_{h}$ satisfies

$\displaystyle\|\sigma-\sigma_{h}\|+\|\mathrm{I}_{h}u-u_{h}\|_{a_{h}}+\|\mathrm{I}_{h}u-u_{h}\|_{\mathrm{s}_{h}}$
$\displaystyle\qquad\lesssim\min_{p_{k+1}\in
P_{k+1}(\mathcal{M})}\|\sigma-\mathbb{C}\varepsilon_{\mathrm{pw}}(p_{k+1})\|+\mathrm{osc}(f,\mathcal{M}),$

where
$\sigma_{h}\coloneqq\mathbb{C}\varepsilon_{\mathrm{pw}}(\mathcal{R}_{h}u_{h})$
and
$\mathrm{osc}(f,\mathcal{M})\coloneqq\|h_{\mathcal{M}}(1-\Pi_{\mathcal{M}}^{k+1})f\|$.
The hidden constant in $\lesssim$ is independent of $\kappa$ and the mesh-
size.

######  Remark 5.7 (alternative eigensolver).

The following alternative approach utilizes an additional reconstruction for
the divergence as suggested in [15]. Given
$v_{h}=(v_{\mathcal{M}},v_{\Sigma})\in V_{h}$, the divergence reconstruction
$\mathrm{div}_{h}v_{h}\in P_{k}(\mathcal{M})$ uniquely solve

$\displaystyle(\mathrm{div}_{h}v_{h},p_{k})_{L^{2}(\Omega)}=-(v_{\mathcal{M}},\nabla_{\mathrm{pw}}p_{k})_{L^{2}(\Omega)}+\sum_{S\in\Sigma}(v_{S}\cdot\nu_{S},[p_{k}]_{S})_{L^{2}(S)}$

for any $p_{k}\in P_{k}(\mathcal{M})$. The discrete problem (1.5) is defined
as in Subsection 5.1 with the following modifications:
$a_{\mathrm{pw}}(u_{\mathrm{nc}},v_{\mathrm{nc}})\coloneqq(\varepsilon_{\mathrm{pw}}(u_{\mathrm{nc}}),\varepsilon_{\mathrm{pw}}(v_{\mathrm{nc}}))_{L^{2}(\Omega)}$
and $a_{h}$ is replaced by

$a_{h}(u_{h},v_{h})\coloneqq 2\mu
a_{\mathrm{pw}}(\mathcal{R}_{h}u_{h},\mathcal{R}_{h}v_{h})_{L^{2}(\Omega)}+\kappa(\mathrm{div}_{h}u_{h},\mathrm{div}_{h}v_{h})_{L^{2}(\Omega)},$

where $\mathcal{R}_{h}$ is the potential reconstruction from (6.1) below,
similar to [15, Section 3.2]. For any $v\in V$, the $a_{\mathrm{pw}}$
orthogonality $v-\mathcal{R}_{h}\mathrm{I}_{h}v\perp W_{\mathrm{nc}}$ [15, Eq.
(19)] implies (1.7). This and the Pythagoras theorem with the $L^{2}$
orthogonality $\mathrm{div}\,v-\mathrm{div}_{h}\mathrm{I}_{h}v\perp
P_{k}(\mathcal{M})$ [15, Proposition 3] shows (A) with

$\displaystyle\|\mathrm{I}_{h}v\|_{a_{h}}^{2}$
$\displaystyle=2\mu\|v\|^{2}_{a_{\mathrm{pw}}}-2\mu\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}+\kappa\|\mathrm{div}v\|^{2}-\kappa\|(1-\Pi_{\mathcal{M}}^{k})\mathrm{div}v\|^{2}$
$\displaystyle\leq
2\mu\|v\|^{2}_{a}-2\mu\|v-\mathrm{G}_{h}v\|_{a_{\mathrm{pw}}}^{2}.$

The condition (B) follows from (5.2); (C) is a consequence of (3.9) and the
Korn inequality (5.5). It turns out that (A)–(C) are valid with the same
constants $\alpha$ and $\beta$ from Corollary 5.5, whence Corollary 5.5 holds
verbatim.

In practice, the computation of the LEB in Corollary 5.5 requires an upper
bound of the Korn constant in convex polyhedra. This is closely related to the
continuity constant of a right-inverse of the divergence operator and is
available in 2d from [12, 20]. For any $K\in\mathcal{M}$, let $z_{K}$ be the
midpoint of a largest ball inscribed in $K$ with nodes $x_{1},\dots,x_{m}$.
Define the geometric parameter

$\displaystyle\varrho(K)\coloneqq\frac{\mathrm{dist}(z_{K},\partial
K)}{\max_{j=1,\dots,m}|z_{K}-x_{j}|}\quad\text{and}\quad\varrho\coloneqq\max_{K\in\mathcal{M}}\varrho(K),$

then [20, Lemma 6.2] provides the upper bound

$\displaystyle
c_{\mathrm{Korn}}\leq\sqrt{1+\frac{4}{\varrho^{2}}(1+\sqrt{1-\varrho^{2}})}$

of the Korn constant. For right-isosceles triangles, $c_{\mathrm{Korn}}\leq
7.318$.

Figure 3. (a) initial triangulation and (b) convergence history plot of
$\lambda_{\mathrm{C}}(1)-\mathrm{LEB}(1)$ for the linear elasticity eigenvalue
problem in Subsection 5.3

###  5.3. Computer benchmark

We approximate the first linear elasticity eigenvalue in the Cook’s membrane
$\Omega\coloneqq\mathrm{conv}\\{(0,0),(48,44),(48,60)$, $(0,44)\\}$ with the
Dirichlet boundary
$\Gamma_{\mathrm{D}}\coloneq\mathrm{conv}\\{(0,0),(0,44)\\}$, Neumann boundary
$\Gamma_{\mathrm{N}}\coloneqq\partial\Omega\setminus\Gamma_{\mathrm{D}}$, and
parameters $\mu=1/2$ and $\kappa=1000$. The initial triangulation of $\Omega$
is displayed in Figure 3(a). The reference value $\lambda(1)=2.9020\times
10^{-4}$ stems from the bound
$\mathrm{LEB}(1)\leq\lambda(1)\leq\lambda_{h}(1)$. We set $\sigma\coloneqq
2^{-1}c_{\mathrm{Korn}}^{-2}(\pi^{-2}+c_{\mathrm{tr}})^{-1}=0.016049$ so that
$\min\\{1,1/(1/2+c_{\mathrm{Korn}}^{2}h_{\mathrm{max}}^{2}\lambda_{h}(j)/\pi^{2})\\}\lambda_{h}(j)\leq\lambda(j)$
from Corollary 5.5. The computation of $\lambda_{\mathrm{C}}(1)$ and
$u_{\mathrm{C}}(1)$ follows Subsection 3.4.4. For the first discrete
eigenvalue $u_{h}(1)=(u_{\mathcal{M}}(1),u_{\Sigma}(1))$, the bound

$\displaystyle\lambda_{\mathrm{C}}(1)-\lambda_{h}(1)\leq\|u_{\mathrm{C}}(1)-\mathcal{R}_{h}u_{h}(1)\|^{2}_{a_{\mathrm{pw}}}+\|u_{h}(1)\|_{\mathrm{s}_{h}}^{2}$

similar to (3.10) motivates a refinement indicator by localization of the
right-hand side. Figure 3(b) displays the convergence history of
$\lambda_{\mathrm{C}}(1)-\mathrm{LEB}(1)$ and the observations of Subsection
3.4.4 apply.

##  6\. Compact embedding in linear elasticity

The quality of lower eigenvalue bounds depends on the accuracy of constants in
local embeddings. For the HHO eigensolvers of this paper however, they also
directly influence the numerical scheme in the choice of $\sigma$ for the
stabilization. An overestimation in trace or Korn inequalities leads to
inaccurate lower eigenvalue bounds on coarse meshes in all computer benchmarks
of the previous three sections. This section proposes eigensolvers to compute
improved guaranteed bounds of the relevant constants in Theorem 2.1, which can
significantly reduce the size of the preasymptotic range. Given a convex
polyhedra $\Omega$ of diameter $d\coloneqq\mathrm{diam}(\Omega)$ and the set
of sides $\mathcal{E}$, define the piecewise constant weight function
$\ell(\partial\Omega)\in P_{0}(\mathcal{E})$ by

$\displaystyle\ell(\partial\Omega)|_{S}=\ell(S,\Omega)\quad\text{for any
}S\in\mathcal{E}$

with $\ell(\Omega,S)$ from (3.2). We seek lower bounds $\gamma_{*}(\Omega)$ of

(6.1) $\displaystyle\gamma(\Omega)=\min_{v\in
V\setminus\\{0\\}}\|v\|_{a}^{2}/\|v\|^{2}_{b}$

in the space $V=\\{v\in H^{1}(\Omega)^{n}:\int_{\Omega}v\,\mathrm{d}x=0\text{
and }\int_{\Omega}\mathrm{D}_{\mathrm{ss}}v\,\mathrm{d}x=0\\}$ with

$\displaystyle
a(v,v)\coloneqq(\varepsilon(v),\varepsilon(v))_{L^{2}(\Omega)}\quad\text{and}\quad
b(v,v)\coloneqq(\ell(\partial\Omega)^{-1}v,v)_{L^{2}(\partial\Omega)}+d^{-2}(v,v)_{L^{2}(\Omega)}.$

Notice that $\gamma(\Omega)^{-1}$ is the best possible constant in
$\|v\|_{b}^{2}\leq\gamma(\Omega)^{-1}\|v\|_{a}^{2}$ for all $v\in V$ and is
utilized in Corollary 5.5 for LEB of the linear elasticity eigenvalue problem.

###  6.1. HHO eigensolver

For the problem (6.1) at hand, set $V_{\mathrm{nc}}\coloneqq
H^{1}(\mathcal{M})^{n}$,
$a_{\mathrm{pw}}(u_{\mathrm{nc}},v_{\mathrm{nc}})\coloneqq(\varepsilon_{\mathrm{pw}}(u_{\mathrm{nc}}),\varepsilon_{\mathrm{pw}}(v_{\mathrm{nc}}))_{L^{2}(\Omega)}$,
and $W_{\mathrm{nc}}\coloneqq P_{k+1}(\mathcal{M})^{n}$. For any
$v_{h}=(v_{\mathcal{M}},v_{\Sigma})\in P_{k+1}(\mathcal{M})^{n}\times
P_{k+1}(\Sigma)^{n}$, the potential reconstruction $\mathcal{R}_{h}v_{h}\in
W_{\mathrm{nc}}$ of $v_{h}$ from [15] satisfies, for any $\varphi_{k+1}\in
W_{\mathrm{nc}}$, that

$\displaystyle a_{\mathrm{pw}}(\mathcal{R}_{h}v_{h},\varphi_{k+1})$ (6.2)
$\displaystyle\quad=-(v_{\mathcal{M}},\mathrm{div}_{\mathrm{pw}}\varepsilon_{\mathrm{pw}}(\varphi_{k+1}))_{L^{2}(\Omega)}+\sum_{S\in\Sigma}(v_{S},[\varepsilon_{\mathrm{pw}}(\varphi_{k+1})]_{S}\nu_{S})_{L^{2}(S)}.$

This defines $\mathcal{R}_{h}v_{h}$ uniquely up to rigid body motions. The
associated degrees of freedom are fixed by (5.3). Given $k\geq 0$, let
$V_{h}\coloneqq\\{v_{h}=(v_{\mathcal{M}},v_{\Sigma})\in
P_{k+1}(\mathcal{M})^{n}\times
P_{k+1}(\Sigma)^{n}:\int_{\Omega}v_{\mathcal{M}}=0\text{ and
}\int_{\Omega}\mathrm{D}_{\mathrm{ss}}\mathcal{R}_{h}v_{h}\,\mathrm{d}x=0\\}$
denote the discrete ansatz space. We claim that
$\mathrm{I}_{h}v\coloneqq(\Pi_{\mathcal{M}}^{k+1}v,\Pi_{\Sigma}^{k+1}v)\in
V_{h}$ for any $v\in V$. In fact, the fixed degrees of freedom in (5.3) imply
$\int_{K}\mathrm{D}_{\mathrm{ss}}\mathcal{R}_{h}\mathrm{I}_{h}v\,\mathrm{d}x=\int_{K}\mathrm{D}_{\mathrm{ss}}v\,\mathrm{d}x$
for any $K\in\mathcal{M}$ in the proof of Lemma 5.3. Since
$\int_{\Omega}\Pi_{\mathcal{M}}^{k+1}v\,\mathrm{d}x=0$ from
$\int_{\Omega}v\,\mathrm{d}x=0$, this shows

$\displaystyle\int_{\Omega}\mathrm{D}_{\mathrm{ss}}\mathcal{R}_{h}\mathrm{I}_{h}v\,\mathrm{d}x=\int_{\Omega}\mathrm{D}_{\mathrm{ss}}v\,\mathrm{d}x=0$

Therefore, $\mathrm{I}_{h}v\in V_{h}$ and so, $\mathrm{I}_{h}$ is an
interpolation from $V$ onto $V_{h}$. The discrete problem (1.5) is defined
with the bilinear forms

$\displaystyle a_{h}(u_{h},v_{h})$ $\displaystyle\coloneqq
a_{\mathrm{pw}}(\mathcal{R}_{h}u_{h},\mathcal{R}_{h}v_{h}),$ $\displaystyle
b_{h}(u_{h},v_{h})$
$\displaystyle\coloneqq(\ell(\partial\Omega)^{-1}u_{\Sigma},v_{\Sigma})_{L^{2}(\partial\Omega)}+d^{-2}(u_{\mathcal{M}},v_{\mathcal{M}})_{L^{2}(\Omega)},$

and the stabilization
$\mathrm{s}_{h}(u_{h},v_{h})\coloneqq\sum_{K\in\mathcal{M}}\mathrm{s}_{K}(u_{h},v_{h})$,

$\displaystyle\mathrm{s}_{K}(u_{h},v_{h})$
$\displaystyle\coloneqq\sigma\int_{K}h_{K}^{-2}(u_{K}-\mathcal{R}_{h}u_{h})\cdot(v_{K}-\mathcal{R}_{h}v_{h})\,\mathrm{d}x$
(6.3)
$\displaystyle\quad+\sigma\sum_{S\in\Sigma(K)}\ell(S,K)^{-1}\int_{S}(u_{S}-\mathcal{R}_{h}u_{h}|_{K})\cdot(v_{S}-\mathcal{R}_{h}v_{h}|_{K})\,\mathrm{d}s$

for any
$u_{h}=(u_{\mathcal{M}},u_{\Sigma}),v_{h}=(v_{\mathcal{M}},v_{\Sigma})\in
V_{h}$. It is straightforward to verify the following result.

######  Lemma 6.1.

The bilinear form $a_{h}+\mathrm{s}_{h}$ is a scalar product in $V_{h}$. In
particular, there are
$\mathrm{dim}(P_{k+1}(\mathcal{M}))+\mathrm{dim}(P_{k+1}(\Sigma(\partial\Omega)))-M$
finite discrete eigenvalues of (1.5), where $M$ is the number of rigid-body
motions in $n$-d. ∎

###  6.2. Lower eigenvalue bounds

The conditions (A)–(C) are verified in this subsection, which lead to LEB in
Corollary 6.2 below. The Galerkin projection $\mathrm{G}_{h}v\in
W_{\mathrm{nc}}$ of $v\in V_{\mathrm{nc}}$ is defined by (1.8) with (5.4).
Although the focus is on the first eigenvalue $\lambda(1)=\gamma(\Omega)$, LEB
can be obtained for other eigenvalues $\lambda(j)$ as well. Recall
$\gamma=\min_{K\in\mathcal{M}}\gamma(K)$ with $\gamma(K)$ from Lemma 5.4.

######  Corollary 6.2 (LEB in local embeddings).

It holds

$\displaystyle\mathrm{LEB}(j)\coloneqq\min\\{1,\gamma/\sigma\\}\lambda_{h}(j)\leq\lambda(j).$

###### Proof.

The verification of (1.7) is similar to Lemma 5.3, cf. also [15]. This
provides (A) even with equality (instead of $\leq$). Given $v\in V$, the
identity

$\displaystyle\mathrm{s}_{K}(\mathrm{I}_{h}v,\mathrm{I}_{h}v)=\sigma\sum_{S\in\Sigma(K)}\ell(S,K)^{-1}\|v-\mathrm{G}_{h}v\|^{2}_{L^{2}(S)}+\sigma
h_{K}^{-2}\|v-\mathrm{G}_{h}v\|_{L^{2}(K)}^{2}$

for any $K\in\mathcal{M}$ from Lemma 5.3 and Lemma 5.4 imply (B) with
$\alpha=\gamma^{-1}\sigma$. Since $\ell(\partial\Omega)$ is piecewise
constant, the Pythagoras theorem with the orthogonality of $L^{2}$ projections
establish

$\displaystyle\|\mathrm{I}_{h}v\|_{b_{h}}^{2}$
$\displaystyle=\sum_{S\in\Sigma(\partial\Omega)}\ell(\partial\Omega)|_{S}^{-1}\|\Pi_{\Sigma}^{k+1}v\|^{2}_{L^{2}(S)}-d^{-2}\|\Pi_{\mathcal{M}}^{k+1}v\|^{2}$
$\displaystyle=\|v\|_{b}^{2}-\sum_{S\in\Sigma(\partial\Omega)}\ell(\partial\Omega)|_{S}^{-1}\|(1-\Pi_{\Sigma}^{k+1})v\|^{2}_{L^{2}(S)}-d^{-2}\|(1-\Pi_{\mathcal{M}}^{k+1})v\|^{2}$
$\displaystyle\geq\|v\|_{b}^{2}-\sum_{S\in\Sigma(\partial\Omega)}\ell(\partial\Omega)|_{S}^{-1}\|v-\mathrm{G}_{h}v\|^{2}_{L^{2}(S)}+d^{-2}\|v-\mathrm{G}_{h}v\|^{2}$

This, (4.4), the Poincaré, and Korn inequality show (C) with
$\beta=c_{\mathrm{Korn}}^{2}(h_{\mathrm{max}}^{2}/\pi^{2}+\beta_{\mathrm{st}})$
with $\beta_{\mathrm{st}}$ from (4.2). The application of Theorem 2.1 yields

$\displaystyle\min\\{1,1/(\gamma^{-1}\sigma+c_{\mathrm{Korn}}^{2}(h_{\mathrm{max}}^{2}/\pi^{2}+\beta_{\mathrm{st}})\lambda_{h}(j))\\}\lambda_{h}(j)\leq\lambda(j).$

Given a positive parameter $t>0$, consider the domain $\Omega_{t}\coloneqq
t\Omega$ with the partition
$\mathcal{M}_{t}\coloneqq\\{tK:K\in\mathcal{M}\\}$. Since the continuous and
discrete Rayleigh quotients are invariant under scaling, the continuous and
discrete eigenvalue in $\Omega_{t}$ coincide with those in $\Omega$. On the
mesh $\mathcal{M}_{t}$, the arguments from above lead to the LEB

$\displaystyle\min\\{1,1/(\gamma^{-1}\sigma+c_{\mathrm{Korn}}^{2}(t^{2}h_{\mathrm{max}}^{2}/\pi^{2}+t\beta_{\mathrm{st}})\lambda_{h}(j))\\}\lambda_{h}(j)\leq\lambda(j)$

due to the invariance of $c_{\mathrm{Korn}}$ and the scaling
$\beta_{\mathrm{st}}\approx h_{\mathrm{max}}$. The limit of this as $t\to 0$
concludes the proof. ∎

# | $\sigma$ | $\gamma_{*}$
---|---|---
1 | 0.0358473 | 0.286114
2 | 0.286114 | 2.2488
3 | 2.2488 | 6.45254
4 | 6.45254 | 7.01185
5 | 7.01185 | 7.02938
Table 1. Lower bounds of $\gamma$ for the reference triangle in Subsection 6.3

###  6.3. Computer benchmark

Lower bounds $\gamma_{*}(\Omega)=\mathrm{LEB}(1)$ of $\gamma(\Omega)$ in the
reference triangle $\Omega=\mathrm{conv}\\{(0,0),(1,0),(0,1)\\}$ are computed
on a fixed mesh $\mathcal{M}$ created by three successive uniform refinements
of the reference triangle, where each refinement divides every cell of
$\mathcal{M}$ into four congruent cells by connecting the mid points on the
three sides. Notice that the constant $\gamma(\Omega)$ is invariant under
translation, scaling, and half rotation. Since every triangle in $\mathcal{M}$
is the image of $\Omega$ under a combination of these transformations,
$\gamma(K)=\gamma(\Omega)$ for all $K\in\mathcal{M}$ in Lemma 5.4. This leads
to

(6.4)
$\displaystyle\gamma_{*}(\Omega)=\min\\{1,\gamma(\Omega)/\sigma\\}\lambda_{h}(1)\leq\gamma(\Omega)$

in Corollary 6.2. We set $\sigma\coloneqq
c_{\mathrm{Korn}}^{-2}(\pi^{-2}+c_{\mathrm{tr}})^{-1}=0.0358473\leq\gamma(\Omega)$
so that $\lambda_{h}(j)\leq\lambda(j)$ in (6.4). From the numerical
experiments in the previous section, only rough bounds can be expected. For
the choice $k=1$, we obtain the lower bound $\gamma_{*}(\Omega)=0.286114$.
(For reference, $\gamma\geq 7$ below.) Since the computed lower bound
$\gamma_{*}(\Omega)$ is larger than the initial bound
$c_{\mathrm{Korn}}^{-2}(\pi^{-2}+c_{\mathrm{tr}})^{-1}=0.0358473$, we repeat
the computation with $\sigma=\gamma_{*}(\Omega)=0.286114$. This process can be
successively repeated by updating $\sigma=\gamma_{*}(\Omega)$ after each
iteration. Table 1 displays the computed lower bounds and provides
$7\leq\gamma(\Omega)$ in the reference triangle.

$6.9898\leq\gamma$

$6.9344\leq\gamma$

$7.11834\leq\gamma$

$5.52756\leq\gamma$

$7.7239\leq\gamma$

$5.56383\leq\gamma$

Figure 4. Lower bounds of $\gamma$ for different triangular shapes in
Subsection 6.4 Figure 5. Improved convergence history plot of
$\lambda_{\mathrm{C}}(1)-\mathrm{LEB}(1)$ for the linear elasticity eigenvalue
problem in Subsection 6.4

###  6.4. A revisit to linear elasticity

In the setting of Subsection 5.3, we compute lower bounds of $\gamma(K)$ for
all possible triangles $K$ that can occur from the newest vertex bisection as
described in Subsection 6.3. For the initial triangulation of the Cook’s
membrane displayed in Figure 3(a), lower bounds needs to be computed in six
different triangles due to invariance of $\gamma(K)$ under scaling,
translation, and half rotation. These triangles with a computed lower bound of
$\gamma(K)$ are displayed in Figure 4. We obtain the improved bound
$5.52\leq\gamma$. (For comparison,
$c_{\mathrm{Korn}}^{-2}(\pi^{-2}+c_{\mathrm{tr}})^{-1}=0.032098$.) Since
$\|v\|_{L^{2}(K)}^{2}\leq\gamma(K)^{-1}h_{K}^{2}\|\varepsilon(v)\|_{L^{2}(K)}^{2}$
for any $v\in V(K)$, $\beta=\gamma^{-1}h_{\mathrm{max}}^{2}$ in (C). The
choice $\sigma=2.76$ in Corollary 5.5 leads to the LEB

$\displaystyle\mathrm{LEB}(j)=\min\\{1,1/(1/2+h_{\mathrm{max}}^{2}\lambda_{h}(j)/5.52)\\}\lambda_{h}(j)\leq\lambda(j)$

for the linear elasticity eigenvalue problem. Figure 5 displays the
convergence history plot of $\lambda_{\mathrm{C}}(1)-\mathrm{LEB}(1)$ with
improved accuracy in comparison to Figure 3.

###  6.5. Conclusions

In all examples, adaptive computation recovers optimal convergence rates of
the lower eigenvalue bounds towards the exact eigenvalue. Accurate upper
bounds of constants related to local embeddings can be computed and improve
the quality of the lower eigenvalue bounds. Additional applications beyond
those presented in this paper include general scalar elliptic operators
(instead of Laplace) and the biharmonic eigenvalue problem in 2d, where the
property (1.7) is possible [17]. The 3d case remains open. Another application
is the Stokes eigenvalue problem in primal form, where the vanishing
divergence is enforced by using the reconstruction $\mathrm{div}_{h}$ from
Remark 5.7 on the discrete level. Since the conditions (B)–(C) arise from
local estimates in practice, the analysis extends to piecewise constant
$\sigma$ for improved handling of variable material parameters, e.g., in
linear elasticity.

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