#  Einstein Gravity from Einstein Action: Counterterms and Covariance

Martin Krššák Electronic address<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Theoretical Physics, Faculty of
Mathematics, Physics and Informatics, Comenius University in Bratislava,
84248, Slovak Republic

###### Abstract

The field equations of general relativity can be derived from the Einstein
action, which is quadratic in connection coefficients, rather than the
standard action involving the Gibbons-Hawking-York term and counterterm. We
show that it is possible to construct a new counterterm directly for the
Einstein action, which removes divergences and naturally introduces a flat
reference spacetime. The total action is then covariant under simultaneous
transformation of both the spacetime and reference tetrads, and argue that
this is analogous to the Gibbons-Hawking action. We then explore different
perspectives arising naturally from different uses of the reference tetrad,
and explore implications of viewing gravity as fundamentally described in
terms of non-covariant connection coefficients.

##  1 Introduction

The standard approach to general relativity [1, 2, 3], which mostly follows
the original Einstein’s path [4, 5], starts with studying the motion of a
free-falling particle and identifying gravity with the spacetime geometry. The
non-tensorial Christoffel symbols are recognized to play the role of the total
inertial and gravitational forces acting on the particle. However, when we
formulate the field equations for the geometry itself, we usually invoke the
principle of covariance and demand that the field equations are tensorial. We
are then lead to consider the Riemann tensor, removing all of the nontensorial
coordinate-dependent behaviour of Christoffel symbols, and we are practically
uniquely led to Einstein field equations.

An interesting situation occurs if we try to derive these field equations from
the variational principle. As is well-known, it was Hilbert who almost
simultaneously with Einstein derived the field equations from the action
principle using what is nowadays known as the (Einstein-)Hilbert action [6],
given by the scalar curvature

${\mathcal{S}}{}_{\text{H}}=\frac{1}{2\kappa}\int_{\mathcal{M}}\sqrt{-g}\,R,$
(1)

where $\kappa=8\pi$ in the natural units. The remarkable property of this
action is that the scalar curvature contains second derivatives of the metric,
and hence one would naively expect fourth-order field equations. However, as
it turns out, the higher derivatives form a non-dynamical total derivative
term and we are left with second-order field equations.

It was recognized only in the 1970s by York [7] and Gibbons and Hawking [8]
that, due to the total derivative term, we can derive the field equations only
if we require variations of both the metric and their normal derivatives to
vanish at the boundary. This generally represents a consistency problem for
the variational problem, and can be solved by adding the so-called Gibbons-
Hawking-York (GHY) boundary term [7, 8]

${\mathcal{S}}{}_{\text{GHY}}=\frac{1}{\kappa}\int_{\partial\mathcal{M}}\sqrt{-\gamma}\mathcal{K},$
(2)

where $\mathcal{K}$ is the extrinsic curvature of the boundary and
$\gamma_{\mu\nu}$ is the induced metric on the boundary. The GHY term (2) is
chosen in a such way that its variation cancels out the variation of the total
derivative term in (1), and then the Einstein field equations can be derived
consistently, i.e., assuming only variations of the metric to vanish at the
boundary [9].

The motivation of Gibbons and Hawking to consider (2) was not just to vary the
action and obtain the field equations, but to find a finite Euclidean
gravitational action solutions for studying black hole thermodynamics using
path integrals, i.e. to evaluate the action on-shell [8, 10]. Therefore,
adding (2) turned out to be crucial since (1) vanishes in vacuum and hence the
GHY term dominates the action.

However, including the GHY term renders the action divergent in the general
case, requiring regularization by adding a counterterm. The total
gravitational action then takes the form

${\mathcal{S}}{}_{\text{grav}}={\mathcal{S}}{}_{\text{H}}+{\mathcal{S}}{}_{\text{GHY}}+{\mathcal{S}}{}_{\text{counter}},$
(3)

and we face the challenge of finding an appropriate counterterm to eliminate
divergences. We are primarily interested in removing IR divergences coming
from the limit $r\rightarrow\infty$, since the Euclidean time becomes
periodic. In asymptotically flat spacetimes, the original proposal by Gibbons
and Hawking [8], known nowadays as the background subtraction, was to add a
counterterm given by the GHY term for the reference or background metric
${}^{0}g_{\mu\nu}$, which is a flat Minkowski metric in which the spacetime
metric is isometrically embedded. The total gravitational action (3) then
depends on both the spacetime and reference metrics and can be written as [8,
11]

${\mathcal{S}}{}_{\text{grav}}(g,{}^{0}g)={\mathcal{S}}{}_{\text{H}}(g)+{\mathcal{S}}{}_{\text{GHY}}(g)+{\mathcal{S}}{}_{\text{counter}}({}^{0}g).$
(4)

The well-known problem is that the isometric embedding of a metric in
Minkowski spacetime does not have to exist, what lead to development of new
regularization methods, including holographic renormalization, where the
counterterm is a local covariant function of the intrinsic geometry of the
boundary [12, 13], and the recent counterterm for null boundaries [14].

Besides the challenge of determining the appropriate counterterm in specific
situations, the presence of the counterterm can be seen as “somewhat awkward”
[15] and naturally leads to questions that has likely perplexed anyone
encountering the full gravitational action (4) for the first time: How come
that in a fully covariant theory that excludes any background structures, the
divergences are removed using the reference/background metric? Moreover, what
is actually being subtracted from the gravitational action?

##  2 Einstein actions

While Einstein originally did not derive the field equations from the action
principle, he did introduce his own action in October 1916 [16], when he
proposed the Lagrangian

${\mathcal{L}}{}_{\text{E}}=\frac{1}{2\kappa}\sqrt{-g}g^{\mu\nu}(\Gamma^{\rho}{}_{\sigma\mu}\Gamma^{\sigma}{}_{\rho\nu}-\Gamma^{\rho}{}_{\mu\nu}\Gamma^{\sigma}{}_{\rho\sigma}),$
(5)

which can be obtained from the Hilbert Lagrangian by separating the
contribution of the second derivative terms into a total derivative term [1,
17, 18]

${\mathcal{L}}{}_{\text{H}}={\mathcal{L}}{}_{\text{E}}+{\mathcal{L}}{}_{\text{tot}}.$
(6)

Varying this Lagrangian with respect to the metric, we do obtain the vacuum
field equations in their potential form [16], where we essentially separate
the terms containing second derivatives of the metric and the terms quadratic
in first derivatives of the metric [19, 20]. The quadratic terms will be given
by [16]

$t^{\mu}{}_{\nu}=\frac{1}{2}\left[{\mathcal{L}}{}_{\text{E}}\delta^{\mu}{}_{\nu}-\frac{\partial{\mathcal{L}}{}_{\text{E}}}{\partial(\partial_{\mu}g^{\rho\sigma})}\partial_{\nu}g^{\rho\sigma}\right],$
(7)

known at the Einstein energy-momentum pseudotensor, from which the
gravitational energy-momentum can be defined. Additionally, other
pseudotensors, such as the symmetric pseudotensor proposed by Landau and
Lifshitz [1], or Weinberg [2], can also be introduced.

The key characteristic shared by all these pseudotensors is that they are not
tensors; they are expressions quadratic in Christoffel symbols, and hence can
always be made to vanish at a point using the Ricci normal coordinates. In
order to find meaningful predictions for the total energy-momentum, these
pseudotensors need to be evaluated in well-behaved coordinate systems, which
usually asymptotically approach the inertial coordinate system [1]. Despite
their non-tensorial dependence on the coordinates, these pseudotensors do
provide correct answer for the gravitational energy-momentum that agrees with
the Hamiltonian method [21, 22].

The tetrad version of (5) was rather accidentally discovered by Einstein in
the attempt to unify gravity and electromagnetism in late 1920s. To this end,
we consider a set of orthonormal vectors $h_{a}$ called the tetrad at each
point of spacetime, with $h^{a}=h^{a}{}_{\mu}dx^{\mu}$ being its inverse and
having components $h^{a}{}_{\mu}$ in some coordinate basis. The tetrads form a
non-coordinate basis and hence the components of the spacetime metric can be
written as

$g_{\mu\nu}=\eta_{ab}h^{a}_{\ \mu}h^{b}_{\ \nu},$ (8)

where $\eta_{ab}=\text{diag}(-1,1,1,1)$, allowing us to use the tetrads
instead of the metric as a fundamental variable. The tetrads $h^{a}{}$ are
independent of coordinates and change under transformation of a non-coordinate
basis as

$h^{a}\rightarrow\Lambda^{a}{}_{b}h^{b},$ (9)

where $\Lambda^{a}{}_{b}$ is a local Lorentz transformation to ensure
preservation of orthonormality.

We can define the coefficients of anholonomy
$f^{c}{}_{ab}=h_{a}{}^{\mu}h_{b}{}^{\nu}(\partial_{\nu}h^{c}{}_{\mu}-\partial_{\mu}h^{c}{}_{\nu})$,
and follow Einstein111 Actually Einstein presented this in a very different
way in terms of the torsion tensor that was supposed to represent both gravity
and electromagnetism in the unified theory [23]. However, as explained in the
recent paper [24], Einstein’s “torsion tensor” is actually just the
coefficient of anholonomy and hence he effectively did what is presented here.
We return to the teleparallel framework and torsion in Section 4. to search
for a Lagrangian quadratic in the coefficients of anholonomy that yields
symmetric field equations. Einstein found such a Lagrangian in 1929 [25]

${\mathcal{L}}{}_{\text{E}}=-\frac{h}{2\kappa}\left(\frac{1}{4}f^{\rho}{}_{\mu\nu}f_{\rho}{}^{\mu\nu}+\frac{1}{2}f^{\rho}{}_{\mu\nu}f^{\nu\mu}{}_{\rho}-f^{\nu\mu}{}_{\nu}f^{\nu}{}_{\mu\nu}\right).$
(10)

While this Lagrangian looks naively very different from both the Hilbert (1)
and Einstein (5) Lagrangians, recalling that the Ricci rotation coefficients
are related to the coefficients of anholonomy through [24]

$\omega^{a}{}_{bc}=\frac{1}{2}\Big{[}f_{b}{}^{a}{}_{c}+f_{c}{}^{a}{}_{b}-f^{a}{}_{bc}\Big{]},$
(11)

we can recast it as

${\mathcal{L}}{}_{\text{E}}=\frac{h}{2\kappa}\left(\omega^{a}_{\,\,\,ca}\omega^{bc}{}_{b}-\omega^{a}{}_{cb}\omega^{bc}{}_{a}\right).$
(12)

From here it should be obvious that (12) is a tetrad version of (5), and hence
Einstein in 1929 indeed “just” rediscovered his earlier 1916 Lagrangian (5) in
tetrad formalism [24].

The alternative approach is to start with the tetrad Hilbert action, i.e. a
scalar curvature in terms of the Ricci rotation coefficients (11), and
separate the total derivative term along the lines of (6), resulting in (12).
This was first done by Møller in 1961 [26], and hence (12) is often referred
as the Møller Lagrangian. It is then possible to define an analogue of the
Einstein pseudotensor known as the Møller complex, which is a coordinate-
tensor but a pseudotensor with respect to (9), from which the total energy-
momentum can be defined as well [27, 28, 29].

##  3 Counterterms for the Einstein action

We can now use the fact that the Hilbert action can be decomposed as (6) in
both the metric and tetrad formulation, and rewrite the total gravitational
action (3) as

${\mathcal{S}}{}_{\text{grav}}={\mathcal{S}}{}_{\text{E}}+{\mathcal{S}}{}_{\text{tot}}+{\mathcal{S}}{}_{\text{GHY}}+{\mathcal{S}}{}_{\text{counter}}.$
(13)

We can then observe that the dynamics is fully contained in the Einstein term
${\mathcal{S}}{}_{\text{E}}$, from which the field equations are derived, and
which is divergent, similar to (4) before adding the counterterm. This brings
us to the idea that the total derivative and GHY terms are not necessary in
the total gravitational action (13), and are rather just remnants of the
mathematical formulation where the Hilbert action (1) is the starting point.
We would like to argue here that it is possible to find a counterterm directly
for the Einstein action, i.e. write the total gravitational action as

${\mathcal{S}}{}_{\text{grav}}={\mathcal{S}}{}_{\text{E}}+{\mathcal{S}}{}_{\text{counter}},$
(14)

where the counterterm is capable to not only regularize the Einstein action,
but also ensure that the total gravitational action (14) is as covariant as
the Gibbons-Hawking gravitational action (4).

To construct the counterterm, we choose to do so for the tetrad (12) Einstein
action, but the construction for the metric Einstein action (5) is analogous.
We use the fact that the Einstein action (12) under a change of the non-
coordinate basis (9), transforms as222This can be obtained from a rather
lengthy but straightforward calculation, or seen directly from the equivalent
teleparallel formulation discussed in Section 4 and using the results of [30].

${\mathcal{L}}{}_{\text{E}}\rightarrow{\mathcal{L}}{}_{\text{E}}+\frac{1}{\kappa}\partial_{\mu}\left[hh_{a}{}^{\nu}h_{d}{}^{\mu}\eta^{bd}\Lambda^{a}_{\
c}\partial_{\nu}(\Lambda^{-1})^{c}_{\ b}\right],$ (15)

and consider a reference tetrad representing a general tetrad in Minkowski
spacetime

${}^{0}h^{a}{}=\Lambda^{a}{}_{b}\tilde{{}^{0}h}{}^{b},$ (16)

where $\tilde{{}^{0}h}{}^{b}$ is the tetrad with components
$\tilde{{}^{0}h}{}^{b}_{\ \mu}=\text{diag}(1,1,1,1)$ in the Cartesian
coordinate system. Since the Riemannian spin connection333The Riemannian spin
connection in the general spacetime is defined from the Ricci rotation
coefficients (11) as $\omega^{a}{}_{b\mu}=\omega^{a}{}_{bc}h^{c}{}_{\mu}$
vanishes for $\tilde{{}^{0}h}{}^{b}$, we find that for the reference tetrad
(16) it can be written as

$\omega^{a}{}_{b\nu}({}^{0}h^{a}{})=\Lambda^{a}_{\
c}\partial_{\nu}(\Lambda^{-1})^{c}_{\ b},$ (17)

which follows from transformation properties of the spin connection [31], and
is precisely the term appearing in (15).

We then choose the counterterm as the total derivative term in (15) expressed
in terms of the reference tetrad (16), i.e.

${\mathcal{L}}{}_{\text{counter}}(h^{a},{}^{0}h^{a})=\frac{1}{\kappa}\partial_{\mu}{\mathcal{B}}{}^{\mu}(h^{a},{}^{0}h^{a})=\frac{1}{\kappa}\partial_{\mu}\left[hh_{a}{}^{\nu}h_{d}{}^{\mu}\eta^{bd}\omega^{a}{}_{b\nu}({}^{0}h^{a}{})\right],$
(18)

and the full gravitational action takes the form

${{\mathcal{S}}{}}_{\text{grav}}(h^{a},{}^{0}h^{a})={\mathcal{S}}{}_{\text{E}}(h^{a})+{\mathcal{S}}{}_{\text{counter}}(h^{a},{}^{0}h^{a}),$
(19)

where $h^{a}$ and ${}^{0}h^{a}$ are the spacetime and reference tetrads,
respectively.

Based on the analogy with the Gibbons-Hawking action (4), we require that the
total action (19) vanishes for the Minkowski spacetime, what is achieved if we
identify the full tetrad with the reference tetrad, i.e. $h^{a}={}^{0}h^{a}$.
In asymptotically flat spacetimes, we require that the spacetime tetrad
reduces to the reference tetrad in the limit $r\rightarrow\infty$, same as in
the case of Gibbons-Hawking action (4).

Let us illustrate this on the example of the Schwarzschild solution. We take
the diagonal tetrad

$h^{a}{}_{\mu}=\text{diag}(f,f^{-1},r,r\sin\theta),\qquad
f^{2}=1-\frac{2M}{r},$ (20)

for which the Einstein action (12) is

${\mathcal{S}}{}_{\text{E}}=\frac{1}{\kappa}\int_{\mathcal{M}}{}\sin\theta=\frac{1}{2}\int
dt\left.r\,\right|_{r_{A}}^{r_{B}},$ (21)

where $r_{A}$ and $r_{B}$ are the bounds of integration. In the case of
asymptotically flat spacetime, $r_{B}\rightarrow\infty$ and the action
diverges regardless of whether we choose $r_{A}$ to be at the origin or the
horizon.

We can then consider the reference tetrad
${}^{0}h^{a}{}_{\mu}=\text{diag}(1,1,r,r\sin\theta)$, and calculate the
counterterm as

${\mathcal{S}}{}_{\text{counter}}=\frac{1}{\kappa}\int_{\mathcal{M}}{}\frac{2M+(f-2)r}{fr}\sin\theta.$
(22)

Adding it to (21) and integrating the spatial part, we obtain

${\mathcal{S}}{}_{\text{grav}}={\mathcal{S}}{}_{\text{E}}+{\mathcal{S}}{}_{\text{counter}}=\int
dt\left.r(1-f)\right|_{r_{A}}^{r_{B}}.$ (23)

However, when performing the integration leading to (23), we must be careful
about the singularity and discontinuity of the Lagrangian at the horizon. We
need to ensure that we integrate only over the manifold patches where the
function is smooth. Therefore, we can consider the action either for the
exterior or the interior solution of the black hole. The exterior solution is
relevant in the Euclidean case, which covers only the exterior region of a
black hole and time becomes periodic with a period $\beta=8\pi M$. We then
choose $r_{A}=2M$ and $r_{B}\rightarrow\infty$, and find the Euclidean action
$S_{\text{grav}}=-i\beta M$, which is exactly twice the value of the original
result by Gibbons and Hawking [8] using (4)444 Note that the same factor 2 was
noticed for the metric Einstein action (5), or rather its teleparallel
formulation discussed in Section 4, in [32]. Moreover, it was suggested to the
use of the so-called canonical frames could resolve the problem of the factor
of 2 [33]. For an alternative resolution, see our upcoming paper [34]..

The interior solution is relevant for the recently proposed complexity=action
duality, where it corresponds to the Wheeler-de Witt (WdW) patch at late times
[35]. In this case, we are primarily interested not in the action itself but
rather its time derivative known as the action growth. Choosing $r_{A}=0$ and
$r_{B}=2M$, we find the action growth at the WdW patch to be
$d{\mathcal{S}}{}_{\text{grav}}/dt=2M$ [36], which exactly agrees with the
standard GHY method for the interior region of a black hole [35]. Moreover,
this agreement for the interior solutions can be found not only in the
Schwarzschild case, but also when the electric charge and negative
cosmological constant are included [36].

##  4 Four faces of the gravitational action

The total gravitational action can be then viewed from four different
perspectives. Besides the standard Gibbons-Hawking action (4), the Einstein
action with the appropriate counterterm (19) is a viable alternative.
Moreover, two further perspectives on the gravitational action can be found by
using the reference tetrad in different ways.

The first option is to effectively eliminate ${}^{0}h^{a}$ and view the theory
within the special class of tetrads. We start with $\\{h^{a},{}^{0}h^{a}\\}$
and consider some local Lorentz transformation $\Lambda^{a}{}_{b}$ that
transforms ${}^{0}h^{a}$ to the Cartesian tetrad $\tilde{{}^{0}h}^{a}$, what
turns the counterterm into zero and the reference tetrad will effectively
disappear from the theory. We then obtain a special class of tetrads for which
the Einstein action (12) will give the correct regularized action without a
counterterm. These are the so-called proper tetrads known from teleparallel
gravity [37, 38, 24].

The advantage of this perspective is that the existence of proper tetrads
explains why the Møller complex evaluated in a special class of tetrads gives
the correct finite prediction for the energy-momentum: the proper tetrads are
the tetrads in which the counterterm (18) is trivial, and the Einstein action
(12) is regular and hence the Møller complex derived from it gives the correct
regularized conserved charges. An analogous argument in the metric formalism
for the original Einstein’s 1916 action (5) explains why the Einstein (7) and
other pseudotensors have to be evaluated within preferred coordinate systems
to achieve sensible finite predictions: these preferred coordinate systems are
the coordinate basis in which the metric analogue of (18) is trivial.

The last option is to move away from the realm of Riemannian geometry and
recast (19) as the action of teleparallel gravity [31, 39, 24]. We just have
to use the fact that (17) can be identified as the teleparallel connection
${\stackrel{{\scriptstyle\bullet}}{{\omega}}}{}^{a}{}_{b\mu}$ utilized in
teleparallel gravity [31, 39], and use it instead of ${}^{0}h^{a}$. The total
action (19) can be then written fully in terms of the Riemannian
$\omega^{a}{}_{b\mu}$ and teleparalel
${\stackrel{{\scriptstyle\bullet}}{{\omega}}}{}^{a}{}_{b\mu}$ connections.
These two connections are related through the Ricci theorem, which states that
the difference between two such connections is necessarily proportional to the
contortion tensor

${\stackrel{{\scriptstyle\bullet}}{{K}}}{}^{a}{}_{b\mu}={\stackrel{{\scriptstyle\bullet}}{{\omega}}}{}^{a}{}_{b\mu}-\omega^{a}{}_{b\mu}.$
(24)

We can then recast the total Lagrangian (19) as [31, 24]

$\displaystyle{\mathcal{L}}{}_{\text{TG}}(h^{a}{}_{\mu},{\stackrel{{\scriptstyle\bullet}}{{\omega}}}{}^{a}{}_{b\mu})=-\frac{h}{2\kappa}\left[{\stackrel{{\scriptstyle\bullet}}{{K}}}{}^{abc}{\stackrel{{\scriptstyle\bullet}}{{K}}}{}_{cba}-{\stackrel{{\scriptstyle\bullet}}{{K}}}{}^{ac}{}_{a}{\stackrel{{\scriptstyle\bullet}}{{K}}}{}^{b}{}_{cb}\right].$
(25)

The equivalence can also be seen from the fact that the resulting total action
in the Schwarzschild case (23) is equivalent to the results obtained within
the teleparallel framework using either working with proper tetrads or
calculating the spin connection [36, 38, 30]. For further discussion and
alternative viewpoints, see [40, 32, 41].

The advantage of the teleparallel form of the action is that (25) is expressed
in terms of the tensorial quantities and is manifestly covariant under the
simultaneous transformation of the tetrad and teleparallel connection.
Therefore, we can view teleparallel geometry as a geometric framework where
the Einstein action (12) is naturally covariantized [24], albeit in a sense
that we discuss in the following section.

##  5 Covariance of the gravitational action

While both the Einstein term (12) and the counterterm (18) are not covariant
under a change of tetrad (9), their combination in the total action (19) is
covariant under a simultaneous transformation of both the tetrad and reference
tetrad

$\\{h^{a},{}^{0}h^{a}\\}\rightarrow\\{\Lambda^{a}{}_{b}\,h^{b},\Lambda^{a}{}_{b}\,{}^{0}h^{b}\\}.$
(26)

Note that this is different from covariance under a change of $h^{a}$ alone,
and is analogous to the situation in the teleparallel framework, where the
action (25) is covariant under simultaneous transformation of both the tetrad
and spin connection. This is often viewed as an analogue of the Stückelberg
trick [42, 32], and is recently a matter of discussion whether this covariance
should be taken seriously or considered artificial [43, 44, 24].

We have constructed the regularized Einstein action (19) in analogy with the
construction of the full gravitational action (4), and hence it should be
obvious that it is as natural/artificial as the full gravitational action of
general relativity (4). Therefore, covariance of (19), or (25), should be
discussed not in comparison with the Hilbert term alone but rather with
respect to the total action including all the boundary terms (4).

While the Hilbert action is covariant with respect to transformations of the
metric/tetrad alone, the covariance of the boundary terms is a rather subtle
issue. The reason is that the GHY term is covariant with respect to
transformation of the boundary metric, which is determined through
$\gamma^{\mu\nu}=g^{\mu\nu}-n^{\mu}n^{\nu}$, where $n^{\mu}$ is a normal
vector to the boundary. Therefore, the GHY term is covariant only under
simultaneous transformation of both $g^{\mu\nu}$ and $n^{\mu}$ in the bulk
spacetime. Indeed, the extrinsic curvature scalar
$\mathcal{K}=\nabla_{\mu}n^{\mu}$ is a scalar only if we transform both the
bulk metric and normal vector simultaneously.

There are then only two options to keep the total gravitational action (4)
covariant. The first is to transform both the spacetime and reference metrics
simultaneously, what is a direct analogue of (26). In this case, a non-
tensorial change of the GHY term under a change of the spacetime metric alone
is counterbalanced by a non-covariant change of the GHY term for the reference
metric. The other viable option is to keep the reference metric fixed, and
transform both the spacetime metric and the normal vector
simultaneously555Although this option is rather unconventional since the
reference metric is not in the same coordinate system as the spacetime metric.
It is then not clear how the reference metric can be properly matched to the
full metric., what ensures the covariance of the GHY term and, hence, the
total action (4). In both cases, the total gravitational action (4) is not
covariant under transformation of the spacetime metric alone, requiring
transformation of some additional structure: either the reference metric or
the normal vector.

##  6 Discussion and conclusions

The connection coefficients play the key role in general relativity where they
represent accelerations measured by the observers. However, due to the
principle of covariance, they are used only through the curvature tensor and
covariant derivatives derived from them, effectively eliminating all of their
non-tensorial behavior from the theory. While this is certainly a reasonable
demand for the field equations, it appears to be incompatible with the action
principle. If we demand a finite action with a well-posed variational
principle, we are forced to supplement the Hilbert term by the GHY term and an
appropriate counterterm (4). However, these terms turn out to depend not only
on the bulk metric but also on foliations and/or the reference metric, which
must be transformed simultaneously with the metric to render the total action
covariant.

We have argued here in favor of reappraisal of the Einstein actions, (5) and
(12), which are quadratic in connection coefficients. These actions do contain
all the dynamics necessary to derive the field equations, but transform by a
surface term under a change of basis that cause divergences. It is then
possible to use this feature and define a counterterm that removes divergences
and makes the total action (19) covariant under simultaneous transformation of
both the spacetime and reference tetrads (26). We have argued that this
analogous to covariance of the Gibbons-Hawking action (4).

Note that the idea of reference/background spacetimes was first introduced by
Rosen, who proposed covariantization of pseudotensors using a reference metric
back in 1940 [45]. The difference with the approach presented here is that the
reference spacetime enters the action only through the total derivative term,
and hence is guaranteed to not change the dynamics. The idea of Rosen is
actually naturally realized in teleparallel framework, where we recognize that
the Riemannian connection of the flat background spacetime has zero curvature
and hence defines a teleparallel connection that can be used to recast the
action into teleparallel framework (25). Indeed, our approach was motivated by
previous results in teleparallel gravity [46, 37, 38, 30, 36].

In this paper, we have decided to start with the well-known Einstein action in
order to demonstrate that analogous construction can be made entirely within
the Riemannian geometry. We built the counterterm based on the analogy with
the standard Gibbons-Hawking term (4). As a result, we were able to show
analogies among the Gibbons-Hawking action (4), Einstein action with the
counterterm (19), and the teleparallel action (25), as well as provide an
explanation why the original Einstein action (12) had to be evaluated in some
privileged tetrad. All these different viewpoints came up naturally depending
on how we used the reference tetrad ${}^{0}h^{a}$.

Our goal was to highlight the fundamental role of connection coefficients in
general relativity and to illustrate that the covariance of the field
equations does not necessarily justify the common assertion that only
covariant objects are meaningful. The covariant field equations can be derived
from the non-covariant Einstein action, and this action, when supplemented by
an appropriate counterterm (19), is as meaningful (and as covariant) as the
Gibbons-Hawking action (4).

Our approach naturally explains the necessity to subtract some reference
structure from the gravitational action. If the gravitational action is
covariant only under simultaneous transformation (26), it is clear that a
reference tetrad or metric must be introduced. This also provides a rather
straightforward explanation of what is being subtracted from the gravitational
action using the reference spacetime. By acknowledging that the fundamental
action is constructed from the connection coefficients, we are merely removing
the effects associated with the non-tensorial character of the connection
coefficients. One can then argue that these are the effects related to the
choice of an observer, i.e., inertial effects.

##  7 Acknowledgements

This work was funded through SASPRO2 project AGE of Gravity: Alternative
Geometries of Gravity, which has received funding from the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie grant agreement No. 945478.

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# An adaptive mesh refinement strategy to ensure quasi-optimality of the
conforming finite element method for the Helmholtz equation via T-coercivity

Tim van Beeck$\,{}^{\text{a}}$ Umberto Zerbinati$\,{}^{\text{b}}$

###### Abstract

It is well known that the quasi-optimality of the Galerkin finite element
method for the Helmholtz equation is dependent on the mesh size and the wave-
number. In literature, different criteria have been proposed to ensure quasi-
optimality. Often these criteria are difficult to obtain and depend on wave-
number explicit regularity estimates. In the present work, we focus on
criteria based on T-coercivity and weak T-coercivity, which highlight mesh
size dependence on the gap between the square of the wavenumber and Laplace
eigenvalues. We also propose an adaptive scheme, coupled with a residual-based
indicator, for optimal mesh generation with minimal degrees of freedom.

###### keywords:

Helmholtz , (weak) T-coercivity , adaptive mesh generation

###### MSC:

65N30 , 65N50 , 65N15

††journal: Computer and Mathematics with Applications

organization=Institute for Numerical and Applied Mathematics, University of
Göttingen, addressline=Lotzestraße 16-18, city=Göttingen, postcode=37073,
country=Germany

organization=Mathematical Institute, University of Oxford, addressline=Andrew
Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, city=Oxford,
postcode=OX2 6GG, country=United Kingdom

##  1 Introduction

In this article, we consider the Helmholtz equation

$-\Delta u-k^{2}u=f\quad\text{in }\Omega,\quad\mathcal{B}u=0\quad\text{on
}\partial\Omega,$ (1)

where $\mathcal{B}$ is an operator encoding the boundary conditions. Our
interest in the Helmholtz equation arises from its wide range of applications
in acoustics, electromagnetics, elastodynamics, and seismology. In this paper,
we are interested in Galerkin discretizations of the Helmholtz equation, i.e.
we seek to find a solution $u_{h}\in V_{h}$ such that

$a(u_{h},v_{h})+b(u_{h},v_{h})=(f,v_{h})_{L^{2}(\Omega)}\quad\forall v_{h}\in
V_{h},$ (2)

where $V_{h}$ is a finite-dimensional space encoding the essential boundary
conditions imposed by $\mathcal{B}$, $a:V_{h}\times V_{h}\to\mathbb{C}$ is the
sesquilinear form

$a(u,v):=(\nabla u,\nabla v)_{L^{2}(\Omega)}-k^{2}(u,v)_{L^{2}(\Omega)},$ (3)

and $b:V_{h}\times V_{h}\to\mathbb{C}$ is a sesquilinear form encoding the
natural boundary conditions imposed by $\mathcal{B}$. We will denote
$\mathcal{A}(u,v)$ the sesquilinear form $a(u,v)+b(u,v)$. It is immediate to
see that the sesquilinear form $a:V_{h}\times V_{h}\to\mathbb{C}$ lacks
coercivity, i.e. there exists no constant $\alpha>0$ such that

$\Re\left\\{a(u_{h},u_{h})\right\\}\geq\alpha\lVert
u_{h}\rVert^{2}_{H^{1}_{k}(\Omega)}\quad\forall u_{h}\in V_{h},$ (4)

where $H^{1}_{k}(\Omega)$ is the Sobolev space $H^{1}(\Omega)$ endowed with
the norm $\lVert u\rVert_{H^{1}_{k}(\Omega)}=\lVert
u\rVert_{H^{1}(\Omega)}+k^{2}\lVert u\rVert_{L^{2}(\Omega)}$. This lack of
coercivity is a well-known issue in the analysis of the Helmholtz equation and
leads to the so-called pollution effect of the wave-number, i.e. the fact that
for a fixed mesh, we lose the quasi-optimality of the Galerkin method as the
wavenumber $k$ increases.

In fact, due to the lack of coercivity of the sesquilinear form $a:V_{h}\times
V_{h}\to\mathbb{C}$ we cannot directly apply the Lax-Milgram theorem to show
the well-posedness of the continuous and discrete problems. The usual approach
to overcome this issue is to resort to a Schatz argument [1], which is based
on the observation that the sesquilinear form $a:V_{h}\times
V_{h}\to\mathbb{C}$ satisfies a Gårding’s inequality of the form

$\Re\left\\{a(u_{h},u_{h})\right\\}+k^{2}\lVert
u_{h}\rVert^{2}_{L^{2}(\Omega)}\geq\mathcal{G}\lVert
u_{h}\rVert^{2}_{H^{1}_{k}(\Omega)}\quad\forall u_{h}\in V_{h},$ (5)

where $\mathcal{G}>0$ is a constant independent of $k$. For most boundary
conditions, the Gårding inequality follows from the Poincaré inequality, [2,
3]. For completeness we now introduce the reader to the classical Schatz
argument following [4], i.e. we consider the solution $z\in H^{1}(\Omega)$ to
the adjoint problem to (2) and observe that

$\displaystyle\lVert u-u_{h}\rVert^{2}_{L^{2}(\Omega)}$
$\displaystyle=\mathcal{A}(u-u_{h},z)=\mathcal{A}(u-u_{h},z-z_{h})$ (6)
$\displaystyle\leq M\lVert u-u_{h}\rVert_{H^{1}_{k}(\Omega)}\lVert
z-z_{h}\rVert_{H^{1}_{k}(\Omega)},$

where $z_{h}\in V_{h}$ is the solution to the discrete adjoint problem to (16)
and $M$ is the continuity constant of the sesquilinear form $\mathcal{A}$.
From an Aubin-Nitsche duality argument, it follows that

$\lVert u-u_{h}\rVert_{L^{2}(\Omega)}\leq M\psi(V_{h})\lVert
u-u_{h}\rVert_{H^{1}_{k}(\Omega)},$ (7)

where $\psi(V_{h})$ is the norm of the residual operator associated with the
projection of the solution to the adjoint problem onto the finite element
space $V_{h}$, i.e.

$\psi(V_{h})=\underset{g\in L^{2}(\Omega)}{\sup}\underset{v_{h}\in
V_{h}}{\inf}\frac{\lVert z_{g}-v_{h}\rVert_{H^{1}_{k}(\Omega)}}{\lVert
g\rVert_{L^{2}(\Omega)}},$ (8)

where $z_{g}\in H^{1}(\Omega)$ is the solution to the continuous adjoint
problem with data $g\in L^{2}(\Omega)$. Once bounded the $L^{2}$-norm of the
error, we can use Gårding’s inequality (5) to bound the $H^{1}$-norm of the
error

$\displaystyle\mathcal{G}\lVert u-u_{h}\rVert^{2}_{H^{1}(\Omega)}$
$\displaystyle\leq\Re\left\\{\mathcal{A}(u-u_{h},u-u_{h})\right\\}+k^{2}\lVert
u-u_{h}\rVert^{2}_{L^{2}(\Omega)}$ (9)
$\displaystyle\leq\Re\left\\{\mathcal{A}(u-u_{h},u-v_{h})\right\\}+k^{2}\lVert
u-u_{h}\rVert^{2}_{L^{2}(\Omega)}$ $\displaystyle\leq M\lVert
u-u_{h}\rVert_{H^{1}_{k}(\Omega)}\lVert u-v_{h}\rVert_{H^{1}_{k}(\Omega)}$
$\displaystyle\quad+k^{2}\psi(V_{h})\lVert
u-u_{h}\rVert_{H^{1}_{k}(\Omega)}^{2}.$

If we impose the following threshold on the norm of the residual operator
associated with the projection of the solution to the adjoint problem onto the
finite element space $V_{h}$:

$\psi(V_{h})\leq\left(\frac{\mathcal{G}}{2k^{2}M^{2}}\right)^{\frac{1}{2}},$
(10)

the Galerkin finite element method is quasi-optimal, i.e.

$\lVert u-u_{h}\rVert_{H^{1}_{k}(\Omega)}\leq
2\frac{M}{\mathcal{G}}\inf_{v_{h}\in V_{h}}\lVert
u-v_{h}\rVert_{H^{1}_{k}(\Omega)},$ (11)

It is important to notice that there is an implicit dependence of the
threshold on the wavenumber $k$ and the mesh size $h$. In fact, assuming
$V_{h}$ is a conforming finite element space of order $p$ we can use the
Bramble-Hilbert lemma to rewrite (10) as:

$C_{\mathcal{I}}h^{q}\lVert
z_{g}\rVert_{H^{s}(\Omega)}\leq\left(\frac{\mathcal{G}}{2k^{2}M^{2}}\right)^{\frac{1}{2}},$
(12)

where $C_{\mathcal{I}}$ is the interpolation constant and $q=\min\\{p,s-1\\}$,
with $s$ positive and smaller or equal then the Sobolev space regularity of
$z_{g}$. Via elliptic regularity results [5], we know that if $\Omega$ is
smooth or convex $z_{g}\in H^{2}(\Omega)$ and the following wave-number
explicit bound on the $H^{2}$-norm of the adjoint solution $z_{g}$ holds:

$\lVert z_{g}\rVert_{H^{2}(\Omega)}\leq(1+kC_{\mathcal{S}})\lVert
g\rVert_{L^{2}(\Omega)}.$ (13)

Combining the above two inequalities we obtain the following bound on the mesh
size $h$:

$h^{q}\leq C\left(2kC_{\mathcal{I}}(1+kC_{\mathcal{S}})\right)^{-1}.$ (14)

For the case of $\mathcal{P}^{1}$ finite elements on smooth domains, we have
$p=1$ and we retrieve the well-known bound $h\leq Ck^{-2}$, [6]. While the
previous bound is only a sufficient condition on the mesh size, it has been
shown that for a fixed polynomial degree $k$, it is not possible to choose a
mesh size $h\geq Ck^{-1}$, [7]. For $\mathcal{P}^{2}$ and higger order finite
elements, Schatz’s argument gives us the bound as for $\mathcal{P}^{1}$ finite
elements because we need $\psi(V_{h})$ to be the norm residual operator with
data taken in $L^{2}(\Omega)$ and therefore we can assume at most $z_{g}\in
H^{2}(\Omega)$.

For this reason, multiple authors have proposed to use $hp$-adaptive finite
element methods. For example [8, 9, 10] have shown the quasi-optimality of
$hp$-adaptive finite element methods under the conditions,

$\frac{kh}{p}<\\!\\!<1,\text{ and }p\geq C_{hp}\log(k).$ (15)

At the heart of their result, as of many others, there are wave-number
explicit bounds on the $H^{p+1}$-norm of the adjoint solution $z_{g}$,
similarly to (13). Such bounds are extremely difficult and technical to
obtain. Furthermore, they usually require strong assumptions on the regularity
of the domain $\Omega$. A notable exception, at least from the point of the
regularity of the domain, is [11].

In the present work, we aim to study a criteria that ensures the quasi-
optimality of the Galerkin finite element method for the Helmholtz equation
similar to the one presented in [12]. The criteria here proposed are based on
the notion of T-coercivity [13, 14, 15, 16] and weak T-coercivity [17],
together with a priori error bounds for the Laplace eigenvalue problem.
Instead, in [12] the authors propose the same criteria but based on the notion
of inf-sup stability. It should come with no surprise that the two criteria
are equivalent, as the inf-sup stability of the sesquilinear form
$\mathcal{A}$ is equivalent to its T-coercivity [14]. The proposed criteria
only require minimal regularity on the domain $\Omega$ and make evident the
dependence of the mesh size on the gap between $k^{2}$ and the Laplace
eigenvalues for the domain $\Omega$.

In this paper, we propose a novel adaptive scheme dependent on the criteria
here presented and a residual-based indicator for the Laplace eigenvalue
problem. Such an adaptive scheme, given a starting mesh, aims to produce the
mesh with the smallest number of degrees of freedom for which the Galerkin
method for the Helmholtz problem is quasi-optimal. This paper is organized as
follows: in Section 2 we introduce the abstract framework for the analysis of
the problem, in particular the notion of T-coercivity and weak T-coercivity.
In Section 3 we show the well-posedness of the continuous problem with
homogeneous Dirichlet and Robin boundary conditions. In Section 4 we extend
these arguments to the discrete cases. In Section 5 we present the novel
adaptive scheme. In Section 6 we present numerical results validating the
proposed criteria and the adaptive scheme.

##  2 Abstract framework

In this section, we provide a brief overview of the abstract framework we will
use to analyze (1). In particular, we introduce the notion of T-coercivity
[13, 14] and its generalization weak T-coercivity [17]. We will also discuss
the application of these notions to study the well-posedness of variational
problems and their approximation with conforming Galerkin methods. In the
following, let $V$ be a Hilbert space and $\mathcal{A}:V\times
V\rightarrow\mathbb{C}$ be a bounded sesquilinear form.

######  Definition 1 (T-coercivity).

The sesquilinear form $\mathcal{A}$ is T-coercive on $V$ if there exists a
constant $\alpha>0$ and a bijective operator $T:V\rightarrow V$ such that

$\mathcal{A}(u,Tu)\geq\alpha\|u\|^{2}_{V}\quad\forall u\in V.$

For $f\in V^{\prime}$, we are concerned with the well-posedness of the
problem:

$\text{find }u\in V\text{ s.t. }\mathcal{A}(u,v)=\langle
f,v\rangle_{V^{\prime}}\quad\forall v\in V.$ (16)

The T-coericvity of $\mathcal{A}$ is equivalent to the well-posedness of (16)
as the following result shows.

######  Theorem 2 (Thm. 1 of [14]).

The sesquilinear form $\mathcal{A}$ is T-coercive on $V$ if and only if
$\mathcal{A}$ satisfies the following inf-sup condition

$\exists\alpha^{\prime}>0:\inf_{u\in V\setminus\\{0\\}}\sup_{v\in
V\setminus\\{0\\}}\frac{|\mathcal{A}(u,v)|}{\|v\|_{V}}\geq\alpha^{\prime}>0,$
(17)

and the only element $v\in V$ satisfying $\mathcal{A}(u,v)=0$ for all $u\in V$
is $w=0$.

We want to consider the approximation of the problem (16) with a conforming
Galerkin scheme. For a finite-dimensional subspace $V_{h}\subset V$, we
consider the restriction of $\mathcal{A}$ to $V_{h}\times V_{h}$. Like the
inf-sup condition, T-coercivity is not automatically inherited onto the
discrete problem. Thus, we introduce the following definition.

######  Definition 3.

The sesquilinear form $\mathcal{A}$ is called uniformly Th-coercive on $V_{h}$
if there exists bijective operators $T_{h}:V_{h}\rightarrow V_{h}$ and
$\alpha^{\ast}>0$ independent of $h$ such that

$\mathcal{A}(u_{h},T_{h}u_{h})\geq\alpha^{\ast}\|u_{h}\|^{2}_{V}\quad\forall
u_{h}\in V_{h}.$ (18)

Analogously to Thm. 2 it can be show that uniform Th-coercivity is equivalent
to an uniform inf-sup condition on $\mathcal{A}$ and therefore we have the
following result, cf. [14, Thm. 2].

######  Theorem 4.

Let $\mathcal{A}$ be uniformly Th-coercive on $V_{h}$. Then, the discrete
problem is well-posed and stable. Further, it holds that

$\|u-u_{h}\|_{V}\leq C_{\text{qo}}\inf_{v_{h}\in V_{h}}\|u-v_{h}\|_{V},$ (19)

where
$C_{\text{qo}}:=1+(\|\mathcal{A}\|_{\mathcal{L}(V,V)}\|T_{h}\|_{\mathcal{L}(V,V)})/\alpha^{\ast}>0$.

As shown in [14, Cor. 1], $\mathcal{A}$ is uniformly Th-coercive on $V_{h}$
for $h$ small enough if there exists operators
$T_{h}\in\mathcal{L}(V_{h},V_{h})$ such that

$\lim_{h\rightarrow 0}\|T_{h}-T\|_{\mathcal{L}(V,V)}=0.$ (20)

However, we will avoid using this criterion in our analysis to be more
explicit in the assumptions on the mesh size $h$. When considering the
Helmholtz problem with Robin boundary conditions, we have to deal with the
additional boundary terms encoded in the sesquilinear form $b(\cdot,\cdot)$.
Therefore, we use the concept of weak T-coercivity [17] which can be
interpreted as the compact perturbation of a T-coercive operator. In this
setting, it is more convenient to work with the linear operator
$A\in\mathcal{L}(V,V)$ induced by a sesquilinear form $\mathcal{A}:V\times
V\rightarrow\mathbb{C}$ via the Riesz-isomorphism.

######  Definition 5 (weak T-coercivity).

Let $A\in\mathcal{L}(V,V)$ be a bounded linear operator. We call $A$ weakly
T-coercive, if there exists a bijective operator $T\in\mathcal{L}(V,V)$ and a
compact operator $K\in\mathcal{L}(V,V)$ such that $T^{\ast}A+K$ is coercive.

With standard arguments, it can be shown that weakly T-coercive operators are
Fredholm with index zero, see e.g. [18, Lem. 1.16], i.e. they are either
injective and surjective or neither injective nor surjective. Thus, we have
the following result.

######  Corollary 6.

Let $A\in\mathcal{L}(V,V)$ be the associated operator of the sesquilinear form
$\mathcal{A}$. If $A$ is weakly T-coercive and injective, then $A$ is
bijective and problem (16) is well-posed.

With arguments similar to (20), the approximation of weakly T-coercive
operators can be studied asymptotically. We refer to [17] for the
T-compatibility criterion and to [19] for a generalization. However, since we
are interested in the explicit dependence on the mesh size $h$ and are only
considering conforming approximations, we avoid the use of these criteria and
instead show the following result.

######  Theorem 7.

Let $A\in\mathcal{L}(V,V)$ be such that $A=B+K$, where $B\in\mathcal{L}(V,V)$
is bijective and $K\in\mathcal{L}(V,V)$ is compact. Suppose further that $A$
is injective and that there exists a bijective
$T_{h}\in\mathcal{L}(V_{h},V_{h})$ such that $B$ is uniformly Th-coercive on
$V_{h}$. Then there exists $h_{0}>0$ such that $A$ is uniformly Th-coercive on
$V_{h}$ for $h<h_{0}$.

###### Proof.

We follow the argumentation in [20, Ex. 26.5] which is based on [21] and make
slight adaptatons for the setting of T-coercivity. Let $R_{h}:V\rightarrow
V_{h}$ be the discrete solution map for $B$, i.e. for all $v\in V$ it holds
that $(B(R_{h}(v)-v),w_{h})_{V}=0$ for all $w_{h}\in V_{h}$. Since $B$ is
uniformly Th-coercive, we have that

$\displaystyle\alpha_{0}\|R_{h}(v)\|_{V}$
$\displaystyle\leq\frac{|(BR_{h}(v),T_{h}R_{h}(v))_{V}|}{\|R_{h}(v)\|_{V}}=\frac{|(Bv,T_{h}R_{h}(v))_{V}|}{\|R_{h}(v)\|_{V}}$
$\displaystyle\leq\|B\|_{\mathcal{L}(V,V)}\|T_{h}\|_{\mathcal{L}(V,V)}\|v\|_{V}.$

Thus, it holds that
$\|R_{h}\|_{\mathcal{L}(V,V)}\leq(\|B\|_{\mathcal{L}(V,V)}\|T_{h}\|_{\mathcal{L}(V,V)}/\alpha_{0})$
and we conclude that

$\displaystyle\|R_{h}(v)-v\|_{V}\leq\left(1+\frac{\|B\|_{\mathcal{L}(V,V)}\|T_{h}\|_{\mathcal{L}(V,V)}}{\alpha_{0}}\right)\inf_{w_{h}\in
V_{h}}\|v-w_{h}\|_{V}.$

Therefore, we have that $R_{h}\rightarrow\operatorname{Id}_{V}$ pointwise as
$h\rightarrow 0$ which implies that $R_{h}\rightarrow\operatorname{Id}_{V}$
uniformly as $h\rightarrow 0$ on compact sets [20, Rem. C.5]. Define
$L:=\operatorname{Id}_{V}+B^{-1}K$ and
$L_{h}:=\operatorname{Id}_{V}+R_{h}B^{-1}K$. Then, we have that
$L_{h}\rightarrow L$ in $\mathcal{L}(V,V)$ due to the uniform convergence of
$R_{h}$ to $\operatorname{Id}_{V}$ on compact sets and the compactness of
$B^{-1}K$. We note that $L$ is bijective and
$L^{1}L_{h}=\operatorname{Id}_{V}-L^{-1}(L-L_{h})$. Taking $h<h_{1}$ small
enough s.t. $\|L^{-1}(L-L_{h})\|_{\mathcal{L}(V,V)}\leq 1/2$ ensures that
$L_{h}$ is invertible with

$L_{h}^{-1}L=\sum_{k\in\mathbb{N}}\left(L^{-1}(L-L_{h})\right)^{k},$ (21)

such that $\|L_{h}^{-1}\|_{\mathcal{L}(V,V)}\leq
2\|L^{-1}\|_{\mathcal{L}(V,V)}=:C_{L}$. Consequently, we have that

$\displaystyle\frac{\alpha_{0}}{C_{L}}\|v_{h}\|^{2}_{V}$
$\displaystyle\leq\alpha_{0}\|L_{h}(v_{h})\|^{2}_{V}\leq|(BL_{h}(v_{h}),T_{h}v_{h})_{V}|$
$\displaystyle\leq|(B(L-L_{h})(v_{h}),T_{h}v_{h})_{V}|+|(Av_{h},T_{h}v_{h})_{V}|$
$\displaystyle\leq\|B\|_{\mathcal{L}(V,V)}\|T_{h}\|_{\mathcal{L}(V,V)}\|L-L_{h}\|_{\mathcal{L}(V,V)}\|v_{h}\|_{V}^{2}+|(Av_{h},T_{h}v_{h})_{V}|,$

where the second to last step follows from
$BL=B(\operatorname{Id}_{V}+B^{-1}K)=A$. For $h<h_{2}$ small enough, the first
term on the right hand side can be bounded by $\alpha_{0}/2C_{L}$ such that
for $h<h_{0}=:\min(h_{1},h_{2})$ it holds that

$\frac{\alpha_{0}}{2C_{L}}\|v_{h}\|^{2}_{V}\leq|(Av_{h},T_{h}v_{h})_{V}|.$
(22)

Thus, $A$ is uniformly Th-coercive. ∎

An application of Thm. 4 immediately yields the following result.

######  Corollary 8.

Assume that $A$ is weakly $T$ coercive and injective, i.e. there exist
bijective operators $T,B\in\mathcal{L}(V,V)$ and $K\in\mathcal{L}(V,V)$
compact such that $T^{\ast}A=B+K$. If there exists a bijective operator
$T_{h}\in\mathcal{L}(V_{h},V_{h})$ such that $(T^{\ast})^{-1}B$ is uniformly
Th-coercive, with constant $\alpha_{0,0}$, then the discrete problem is well-
posed and stable. Furthermore, there holds the estimate

$\|u-u_{h}\|_{V}\leq\left(1+\frac{\|A\|_{\mathcal{L}(V,V)}\|T_{h}\|_{\mathcal{L}(V,V)}}{\alpha_{0,0}}\right)\inf_{v_{h}\in
V_{h}}\|u-v_{h}\|_{V}.$ (23)

##  3 Well-posedness of the continuous problem

We consider the well-posedness of the continuous problem of the Helmholtz
equation with homogeneous Dirichlet and Robin boundary conditions. Depending
on the boundary conditions, we either consider the choices
$V=H^{1}_{0}(\Omega)$ or $V=H^{1}(\Omega)$. Let
$(\lambda^{(i)},e^{(i)})_{i\in\mathbb{N}}$ be the eigenpairs of the Laplacian
ordered by increasing value of $\lambda^{(i)}$ and counted with their
algebraic multiplicity. It is well known that the compactness of the embedding
$H^{s}(\Omega)\hookrightarrow H^{1}(\Omega)$, for any $s>\frac{3}{2}$ together
with elliptic regularity ensures also the compactness of the solution operator

$S:H^{1}(\Omega)\\!\rightarrow\\!H^{1}(\Omega),\ f\mapsto u\text{ : }(\nabla
u,\nabla v)_{L^{2}(\Omega)}=(f,v)_{L^{2}(\Omega)}\ \forall v\in
H^{1}_{0}(\Omega).$ (24)

In particular, since $S:H^{1}(\Omega)\to H^{1}(\Omega)$ is a compact self-
adjoint operator we know that the eigenfunctions
$\\{e^{(i)}\\}_{i\in\mathbb{N}}$ form a basis of $H^{1}(\Omega)$, [2, 22].
Without loss of generality, we assume that the eigenfunctions are normalized
such that they have unit $H^{1}$-norm. Since the eigenfunctions form a basis
of $H^{1}(\Omega)$, we can represent any function $u\in V$ as
$u=\sum_{i\in\mathbb{N}}u^{(i)}e^{(i)}$ with coefficients
$u^{(i)}\in\mathbb{C}$. We will denote from now on
$i_{\ast}:=\max\\{i\in\mathbb{N}:\lambda^{(i)}<k^{2}\\}$, i.e. $i_{\ast}$ the
largest index such that $\lambda_{i_{\ast}}$ is smaller than $k^{2}$. At this
point, we can limit our study to the case $i_{\ast}\geq 0$, otherwise the
problem is coercive and Lax-Milgram applies. We define the subspace $W\subset
V$ and the map $T:V\rightarrow V$ as follows:

$W\coloneqq\text{span}_{0\leq i\leq i_{\ast}}\\{e^{(i)}\\}\subset V,\qquad
T\coloneqq\operatorname{Id}_{V}-2P_{W},$ (25)

where $P_{W}\in\mathcal{L}(V,W)$ is the orthogonal projection onto $W$. Note
that $T$ is self-inverse and therefore bijective and acts on the
eigenfunctions as

$Te^{(i)}:=\begin{cases}-e^{(i)}&\text{if }0\leq i\leq i_{\ast},\\\
+e^{(i)}&\text{if }i>i_{\ast}.\end{cases}$ (26)

###  3.1 Homogeneous Dirichlet boundary conditions

First we consider the Helmholtz equation with homogeneous Dirichlet boundary
conditions, i.e. we look for $u\in V=H^{1}_{0}(\Omega)$ such that

$\mathcal{A}_{0}(u,v)\coloneqq a(u,v)=(f,v)_{L^{2}(\Omega)}\quad\forall v\in
H^{1}_{0}(\Omega),$ (27)

where $a(\cdot,\cdot)$ is defined by (3).

######  Lemma 9 (Prop. 1 of [14]).

The sesquilinear form $\mathcal{A}_{0}$ defined in (27) is T-coercive on $V$.

###### Proof.

By definition, $T$ swaps the sign of the eigenfunctions corresponding to
eigenvalues smaller than $k^{2}$. Therefore, we have that

$\displaystyle\mathcal{A}_{0}(u,Tu)$ $\displaystyle:=\sum_{0\leq i\leq
i_{\ast}}\left(\frac{k^{2}-\lambda^{(i)}}{1+\lambda^{(i)}}\right)(u^{(i)})^{2}+\sum_{i>i_{\ast}}\left(\frac{\lambda^{(i)}-k^{2}}{1+\lambda^{(i)}}\right)(u^{(i)})^{2}$
$\displaystyle\geq\alpha\sum_{i\in\mathbb{N}}\lambda^{(i)}(u^{(i)})^{2}\geq\alpha\|u\|^{2}_{H^{1}(\Omega)}\quad\forall
u\in V,$

where $\alpha:=\min_{i\geq
0}\left\\{\left|\frac{\lambda^{(i)}-k^{2}}{1+\lambda^{(i)}}\right|\right\\}>0$.
∎

Thus, an application of Thm. 2 immediately yields the well-posedness of (27)
for $k^{2}\not\in\\{\lambda_{i}\\}_{i\in\mathbb{N}}$.

###  3.2 Robin boundary conditions

Next, we consider the Helmholtz problem with Robin boundary conditions, i.e.
we look for $u\in V=H^{1}(\Omega)$ such that

$\mathcal{A}(u,v)=a(u,v)+b(u,v)=(f,v)_{L^{2}(\Omega)}\quad\forall v\in
H^{1}(\Omega),$ (28)

with the sesquilinear form $a:V\times V\to\mathbb{C}$ defined in (3) and the
sesquilinear form $b:V\times V\to\mathbb{C}$ defined as

$b(u,v)\coloneqq-ik(\gamma_{0}u,\gamma_{0}v)_{L^{2}(\partial\Omega)},$ (29)

where $\gamma_{0}:H^{1}(\Omega)\to L^{2}(\partial\Omega)$ is the trace
operator. We will now show that the operator $A$ associated with the
sesquilinear form $\mathcal{A}$ is weakly T-coercive on $V$. In preparation,
we recall the following result, see e.g. [23, Prop. 8.3].

######  Lemma 10.

Let $\Omega$ be a bounded Lipschitz domain. Then, the trace operator
$\gamma_{0}:H^{1}(\Omega)\rightarrow L^{2}(\partial\Omega)$ is compact.

We define the operator $K\in\mathcal{L}(V,V)$ as the operator corresponding to
the sesquilinear form $b$, i.e.
$(Ku,v)_{H^{1}(\Omega)}:=b(u,v)=-ik(\gamma_{0}u,\gamma_{0}v)_{L^{2}(\partial\Omega)}$.

######  Corollary 11.

Let $\Omega$ be a bounded Lipschitz domain. Then, the operator
$K\in\mathcal{L}(V,V)$ is compact.

###### Proof.

We follow [23, Sec. 8.3]. Let $u\in H^{1}(\Omega)$. We estimate

$\displaystyle\|Ku\|_{H^{1}(\Omega)}$ $\displaystyle=\\!\\!\sup_{v\in
H^{1}(\Omega)\setminus\\{0\\}}\frac{|(Ku,v)_{H^{1}(\Omega)}|}{\|v\|_{H^{1}(\Omega)}}\leq\\!\\!\sup_{v\in
H^{1}(\Omega)\setminus\\{0\\}}\frac{|k|\|\gamma_{0}u\|_{L^{2}(\partial\Omega)}\|\gamma_{0}v\|_{L^{2}(\partial\Omega)}}{\|v\|_{H^{1}(\Omega)}}$
$\displaystyle\leq C_{\gamma_{0}}|k|\|\gamma_{0}u\|_{L^{2}(\partial\Omega)},$

where $C_{\gamma_{0}}$ is the continuity constant of $\gamma_{0}$. We recall
that an operator $K$ is compact, if and only if $u_{n}\rightharpoonup u$
implies that $Ku_{n}\rightarrow Ku$ [23, Prop. 7.2]. Thus, if
$u_{n}\rightharpoonup u$ in $H^{1}(\Omega)$, it follows that from the previous
lemma that $\gamma_{0}u_{n}\rightarrow\gamma_{0}u$ in $L^{2}(\Omega)$ and
therefore $Ku_{n}\rightarrow Ku$ in $H^{1}(\Omega)$, i.e. $K$ is compact. ∎

######  Lemma 12.

The operator $A\in\mathcal{L}(V,V)$ associated with the sesquilinear form
$\mathcal{A}$ is weakly T-coercive on $V$.

###### Proof.

Let $T\in\mathcal{L}(V,V)$ be defined as in (26) and $A_{0}$ be the operator
associated with the sesquilinear form $A_{0}$ defined by (27) on $H^{1}$
instead of $H^{1}_{0}$. Then, we have that
$T^{\ast}A=T^{\ast}A_{0}+T^{\ast}K$, where $T^{\ast}A_{0}$ is coercive with
the same argumentation as in Lemma 9 and $T^{\ast}K$ is compact since $K$ is
compact. ∎

Thus, we conclude that the operator $A$ is Fredholm with index zero and
therefore the Helmholtz equation with Robin boundary conditions has a unique
solution if and only if $A$ is injective. The latter is a well-known result,
but we provide a sketch of the proof for completeness.

######  Lemma 13.

The operator $A\in\mathcal{L}(V,V)$ associated with the sesquilinear form
$\mathcal{A}$ is bijective.

###### Proof.

Since $A$ is weakly T-coercive, it suffices to show that $A$ is injective. Let
$v\in\ker A$, i.e. $\mathcal{A}(v,w)=0$ for all $w\in V$. Then, we have that

$0=|\mathcal{A}(v,v)|\geq-\Im\mathcal{A}(v,v)=k\|\gamma_{0}v\|^{2}_{L^{2}(\partial\Omega)}\Rightarrow\gamma_{0}v=0.$

Now, let $B$ be a ball centered at zero such that $\Omega\subset B$ and let
$\tilde{v}\in H^{1}(B)$ be the extension by zero of $v$ over $\Omega^{c}\cap
B$. It holds that $\tilde{v}\in H^{1}_{0}(B)$ and due to $\frac{\partial
v}{\partial n}|_{\partial\Omega}=0$, we have that $\nabla\tilde{v}\in
H(\operatorname{div},B)$. Therefore, we have that $\Delta\tilde{v}\in
L^{2}(B)$ and since $\tilde{v}=0$ on $\Omega^{c}\cap B$ the unique
continuation principle [20, Thm. 31.4] implies that $\tilde{v}=0$ in $B$.
Thus, we have that $v=0$ in $\Omega$ and therefore $A$ is injective. For more
details we refer to [20, Thm. 35.5]. ∎

######  Remark 14 (Mixed boundary conditions).

With the same techniques, we could also treat mixed boundary conditions of the
form

$-\Delta u-k^{2}u=f\quad\text{in }\Omega,\quad u=0\quad\text{on
}\partial\Omega_{D},\quad\frac{\partial u}{\partial n}-iku=g\quad\text{on
}\partial\Omega_{R},$ (30)

where $\\{\partial\Omega_{D},\partial\Omega_{R}\\}$ is a partition of
$\partial\Omega$.

##  4 $H^{1}$-conforming finite element discretizations

We now consider $H^{1}$-conforming finite element approximations of the
Helmholtz problem with homogeneous Dirichlet and Robin boundary conditions. In
the following, we assume that $\Omega$ is a polygonal domain. Let
$\\{\mathcal{T}_{h}\\}_{h}$ be a sequence of shape regular triangulations of
$\Omega$ and $p\geq 1$. Then, we define $V_{h}\subset V$ to be the
$H^{1}$-conforming finite element space of order $p$, i.e.

$V_{h}:=\\{v\in L^{2}(\Omega):v|_{T}\in\mathcal{P}^{p}(T)\quad\forall
T\in\mathcal{T}_{h}\\}\cap V.$

Let $(\lambda^{(i)}_{h},e^{(i)}_{h})_{i\in\mathbb{N}}$ be conforming
approximations of the eigenpairs $(\lambda^{(i)},e^{(i)})_{i\in\mathbb{N}}$.
As in the previous section, we set

$W_{h}:=\text{span}_{0\leq i\leq i_{\ast}}\\{e^{(i)}_{h}\\}\subset V_{h},$
(31)

and define a bijective operator $T_{h}:V_{h}\rightarrow V_{h}$ through
$T_{h}:=\operatorname{Id}_{V_{h}}-2P_{W_{h}}$, where
$P_{W_{h}}\in\mathcal{L}(V_{h},W_{h})$ is the orthogonal projection onto
$W_{h}$. Note that $T_{h}$ acts on eigenfunctions as

$T_{h}e^{(i)}_{h}:=\begin{cases}-e^{(i)}_{h}&\text{if }0\leq i\leq
i_{\ast},\\\ +e^{(i)}_{h}&\text{if }i>i_{\ast}.\end{cases}$ (32)

We now proceed to analyze the approximations of the Helmholtz problem with
homogeneous Dirichlet and Robin boundary conditions. To this end, we consider
the restrictions of the sesquilinear forms $\mathcal{A}_{0}$ defined by (27)
and $\mathcal{A}$ defined by (28) to $V_{h}\times V_{h}$. We want to study the
problem of finding $u_{h}\in V_{h}$ such that

$\mathcal{A}_{\ast}(u_{h},v_{h})=(f,v_{h})_{L^{2}(\Omega)}\quad\forall
v_{h}\in V_{h},$ (33)

where $\mathcal{A}_{\ast}\in\\{\mathcal{A}_{0},\mathcal{A}\\}$. In view of
Thm. 4 and Thm. 7 is suffices to show that the sesquilinear form
$\mathcal{A}_{0}$ is uniformly Th-coercive on $V_{h}$. We note that it is
clear from [14] that this will indeed be the case asymptotically, i.e. for
$h\rightarrow 0$. However, we want to provide a more explicit condition on the
mesh size $h$.

######  Theorem 15.

Let $C_{\Omega,V}=C_{P}/\kappa$, where $\kappa$ is the coercivity constant of
$-\Delta$ on $V\in\\{H^{1}_{0}(\Omega),H^{1}(\Omega)\\}$ and $C_{P}$ be the
Poincaré constant. Assume that

$h^{2q}<\frac{k^{2}-\lambda^{(i_{\ast})}}{4\sqrt{i_{\ast}}\lambda^{(i_{\ast})}C_{\Omega,V}C_{\mathcal{I}}},$
(34)

with $q\coloneqq\min\\{p,s-1\\}$, where $s$ is the Sobolev index granted by
elliptic regularity for an $H^{1}(\Omega)$-data. Then, the sesquilinear form
$\mathcal{A}_{0}$ defined in (27) is uniformly $T_{h}$-coercive on $V_{h}$.

###### Proof.

Let $u_{h}\in V_{h}$. Using the discrete eigenbasis, we write
$u_{h}=\sum_{i\in\mathbb{N}}u^{(i)}_{h}e^{(i)}_{h}$. As in the continous case,
we have that

$\displaystyle\mathcal{A}_{0}(u_{h},T_{h}u_{h})$ $\displaystyle=\sum_{0\leq
i\leq
i_{\ast}}\left(\frac{k^{2}-\lambda^{(i)}_{h}}{1+\lambda^{(i)}_{h}}\right)(u^{(i)}_{h})^{2}+\sum_{i>i_{\ast}}\left(\frac{\lambda^{(i)}_{h}-k^{2}}{1+\lambda^{(i)}_{h}}\right)(u^{(i)}_{h})^{2}.$

Thus $\mathcal{A}_{0}$ is Th-coercive with constant
$\alpha^{\ast}:=\min_{i\geq
0}\left\\{\left|\frac{\lambda^{(i)}_{h}-k^{2}}{1+\lambda^{(i)}_{h}}\right|\right\\}$
if and only if $\lambda_{h}^{(i_{\ast})}<k^{2}$, since this condition implies
in particular that $\lambda^{(i)}_{h}<k^{2}$ for all $0\leq i<i_{\ast}$, see
also Fig. 2. We also note that for a conforming approximation, we always
approximate the eigenvalues from above. Thus, we have to ensure that

$\displaystyle\lambda_{h}^{(i_{\ast})}-\lambda^{(i_{\ast})}<k^{2}-\lambda^{(i_{\ast})}$

For the approximation of the eigenvalues, the following estimate holds [20,
Thm. 47.10]

$\displaystyle\lambda_{h}^{(i_{\ast})}-\lambda^{(i_{\ast})}$
$\displaystyle\leq\lambda^{(i_{\ast})}4\sqrt{i_{\ast}}C_{\Omega,V}\max_{v\in
S_{m}}\min_{v_{h}\in V_{h}}\|v-v_{h}\|^{2}_{V}$
$\displaystyle\leq\lambda^{(i_{\ast})}4\sqrt{i_{\ast}}C_{\Omega,V}C_{\mathcal{I}}h^{2q},$

where $C_{\Omega,V}=C_{P}/\kappa$ and $C_{P}$ is the Poincaré constant and
$\kappa$ the coercivity constant of $-\Delta$ on $V$. Thus, we want to ensure
that

$\displaystyle\lambda^{(i_{\ast})}4\sqrt{i_{\ast}}C_{\Omega,V}C_{\mathcal{I}}h^{2q}<k^{2}-\lambda^{(i_{\ast})}\Longleftrightarrow\quad
h^{2q}<\frac{k^{2}-\lambda^{(i_{\ast})}}{4\sqrt{i_{\ast}}\lambda^{(i_{\ast})}C_{\Omega,V}C_{\mathcal{I}}}.$

∎

######  Remark 16.

For the Dirichlet case where $V=H^{1}_{0}(\Omega)$, we have that
$C_{\Omega,V}=C_{P}(1+C_{P}^{2})$.

######  Remark 17.

At first glance, one might think that the estimate (34) on the maximal mesh
size sufficient to ensure quasi-optimality would contradict the known
necessary and sufficient conditions presented in [6, 7]. However, Figure 1
shows that the condition (34) is indeed compatible with the condition
presented in [6], i.e. $hk^{2}=\mathcal{O}(1)$, for $\mathcal{P}^{1}$
elements. We also see that the condition (34) agrees with the condition
presented in [24] for $\mathcal{P}^{2}$ elements.

$10^{3}$$10^{4}$$10^{5}$$10^{-16}$$10^{-13}$$10^{-10}$$10^{-7}$$10^{-4}$$\\!\\!\mathcal{O}(k^{-2})$$\\!\\!\mathcal{O}(k^{-\frac{3}{2}})$$\\!\\!\mathcal{O}(k^{-1})$$k$$h$Quasi–optimality$p=1$$p=2$
Figure 1: The figure shows the mesh size that is sufficient to ensure quasi-
optimality determined via (34) for increasing value of $k$ on the unit square.

$\lambda^{(i_{\ast})}$$k^{2}$$\lambda^{(i_{\ast})}_{h}$$h$ small enough Figure
2: We achieve coercivity by choosing $h$ small enough such that
$\lambda^{(i_{\ast})}_{h}<k^{2}$.

For the Dirichlet problem, we can immediately apply Thm. 4 to conclude the
well-posedness and stability of the discrete problem (33). Furthermore, with
standard interpolation results, we directly obtain the following convergence
result.

######  Corollary 18.

Let $u\in H^{1}_{0}(\Omega)\cap H^{s}(\Omega)$, $s>1$, be the unique solution
to (27). If (34) is satisfied, the discrete problem (33) for
$\mathcal{A}_{\ast}=\mathcal{A}_{0}$ has a unique solution $u_{h}\in V_{h}$
and there holds the estimate

$\|u-u_{h}\|_{H^{1}(\Omega)}\leq C_{\text{qo}}\inf_{v_{h}\in
V_{h}}\|u-v_{h}\|_{H^{1}(\Omega)}\leq
C_{\text{qo}}C_{\mathcal{I}}h^{\min\\{p,s-1\\}}\|u\|_{H^{s}(\Omega)},$

with constants $C_{\text{qo}}$ and $C_{\mathcal{I}}$ independent of $h$.

For the Robin case, we obtain a similar result by combining Thm. 7 with Lemmas
12 and 13.

######  Corollary 19.

Let $u\in H^{1}(\Omega)\cap H^{s}(\Omega)$, $s>1$, be the unique solution to
(27). Assume that (34) is satisfied. Then, there exists $h_{0}>0$ such that
for $h<h_{0}$ the discrete problem (33) with $\mathcal{A}_{\ast}=\mathcal{A}$
has a unique solution $u_{h}\in V_{h}$ and there holds the estimate

$\|u-u_{h}\|_{H^{1}(\Omega)}\leq C_{\text{qo}}\inf_{v_{h}\in
V_{h}}\|u-v_{h}\|_{H^{1}(\Omega)}\leq
C_{\text{qo}}C_{\mathcal{I}}h^{\min\\{p,s-1\\}}\|u\|_{H^{s}(\Omega)},$

with constants $C_{\text{qo}}$ and $C_{\mathcal{I}}$ independent of $h$.

###### Proof.

In Lemma 12, we have shown that $T^{\ast}A=T^{\ast}A_{0}+T^{\ast}K$, where
$A_{0}$ is the operator associated with the sesquilinear form
$\mathcal{A}_{0}$ and $K$ is compact. Since $A$ is injective by Lemma 13 and
$A_{0}=(T^{\ast})^{-1}T^{\ast}A_{0}$ is uniformly Th-coercive by Thm. 15, the
statement follows from corollary 8 and standard interpolation results. ∎

##  5 Adaptive scheme

From the previous analysis, we can easily infer an adaptive scheme to generate
a mesh that guarantees the well-posedness and stability of the discrete
problem. The scheme consists of the following steps:
_Step 1: Determine the index $i_{\ast}$._ First, we want to determine the
index $i_{\ast}:=\max\\{i\in\mathbb{N}:\lambda^{(i)}<k^{2}\\}$. If we are on a
simple and well-studied domain like the unit-square, we know the eigenvalues
of the Laplacian explicitly and can determine $i_{\ast}$ directly. Otherwise,
we have to approximate the eigenvalues numerically. In this case, we have to
ensure that the eigenvalues are approximated well enough such that the index
$i_{\ast}$ is determined correctly. It is worth pointing out that any
numerical scheme can be chosen for this step, even one that is different from
the way we will then discretize the Helmholtz equation. Thus, it is
recommended to use a fine mesh and a higher polynomial degree.
_Step 2: Adaptively solving the Laplace eigenvalue problem._ The next step is
to solve the Laplace eigenvalue problem on a sequence of refined meshes until
$k^{2}-\lambda_{h}^{(i_{\ast})}<0$. This condition ensures that the
sesquilinear form is uniformly $T_{h}$-coercive. We note that it is crucial to
perform this step with the same discretization intended for the Helmholtz
problem in the next step.
_Step 3: Solve the Helmholtz problem._ Finally, we solve the Helmholtz problem
on the mesh generated in Step 2.

######  Remark 20 (Modal analysis).

One might think that the first step in our adaptive scheme would be extremely
expensive in comparison to the other steps. This might be the case at a low
wave number, but since we only need to resolve for the index $i_{\ast}$, at a
high wave number the first step is not as expensive as one might think. In
fact, the mesh size required to resolve for the index $i_{\ast}$ can be
coarses than the mesh required for the Helmholtz problem. It is also worth
pointing out that the first step is a one-time cost and that in most practical
applications, modal analysis is a standard preprocessing procedure to ensure
we are considering a wave number for which the Helmholtz equation is well-
posedness. Furthermore, it is much easier to construct fast and efficient
solvers for the Laplace eigenvalue problem than for the Helmholtz problem,
especially in the high-frequency regime. We redirect the reader interested in
fast solvers for the Laplace eigenvalue problem to [25].

######  Remark 21 (Non-conforming discretizations).

The analysis in the previous section and the adaptive scheme can be extended
to non-conforming discretizations. In [18], a discontinuous Galerkin
discretization of (27) has been analyzed using T-coercivity. Note, however,
that one has to be careful because for non-conforming discretization of the
Laplace eigenvalue it is not ensured that the eigenvalues are approximated
from above [22]. Thus, one has to ensure that
$\lambda^{(i_{\ast})}_{h}<k^{2}<\lambda^{(i_{\ast}+1)}_{h}$.

######  Remark 22.

(Adaptivity$\,{}^{2}$) In step 2, we solve the Laplace eigenvalue problem on a
sequence of refined meshes. We would like to point out that it is possible to
use a residual-based error estimator to determine where to refine the mesh for
the subsequent solve. It is important to notice that we are not interested in
adapting the mesh to obtain optimal convergence rates for a single
eigenfunction but rather to ensure that all the first $i_{\ast}$
eigenfunctions are well approximated. To achieve this result we use the
Babuška–Rheinboldt estimator, averged over the first $i_{\ast}+\ell$
eigenfunctions i.e.

$\eta=i_{\ast}^{-1}\sum_{i=1}^{i_{\ast}+\ell}\\!\sum_{K\in\mathcal{T}_{h}}\\!\\!\left(h^{2}_{K}\lVert\Delta
e_{h}^{(i)}+\lambda_{h}^{(i)}e_{h}^{(i)}\rVert_{L^{2}(K)}^{2}+\frac{h_{K}}{2}\lVert\nabla
e_{h}^{(i)}\cdot n\rVert_{L^{2}(\partial
K\backslash\partial\Omega)}^{2}\right),$ (35)

where $\ell$ is a small integer, and $e_{h}^{(i)}$ and $\lambda_{h}^{(i)}$ are
the $i$-th eigenfunction and eigenvalue, $h_{K}$ is the diameter of the
element $K$, and $n$ is the outward unit normal to $\partial K$.

##  6 Numerical experiments

In this section, we present numerical experiments to validate the theoretical
results introduced in this paper. All the geometries and meshes used in the
numerical experiments are generated using the open-source software Netgen [26]
and the finite element computations are performed using the open-source
software Firedrake [27].

###  6.1 Unit-square

As a first example, we consider the unit-square
$\Omega=[0,1]^{2}\subset\mathbb{R}^{2}$. On this geometry, the eigenvalues of
the Laplacian are given by

$\lambda_{i,j}=\pi^{2}(i^{2}+j^{2}),\quad i,j\in\mathbb{N}.$

Since we know the exact eigenvalues we can determine the index $i_{\ast}$
directly. In Figure 3 and Figure 4 we show that the condition
$\lambda_{h}^{i_{\ast}}<k^{2}$ is indeed sufficient to ensure quasi-optimal
convergence rates for both the Dirichlet and Robin boundary conditions.

$10^{-2}$$10^{-1}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{4}$$10^{5}$$\\!\\!\mathcal{O}(h^{2})$$\\!\\!\mathcal{O}(h^{3})$$h$$\lVert
u-u_{h}\rVert_{L^{2}(\Omega)}$$k=10$$p=1$$p=2$$10^{-1.6}$$10^{-1.4}$$10^{-1.2}$$10^{-1}$$10^{-0.8}$$10^{-0.6}$$10^{-0.4}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$\\!\\!\mathcal{O}(h^{2})$$\\!\\!\mathcal{O}(h^{3})$$h$$k=12$$p=1$$p=2$$10^{-1.6}$$10^{-1.4}$$10^{-1.2}$$10^{-1}$$10^{-0.8}$$10^{-0.6}$$10^{-0.4}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$\\!\\!\mathcal{O}(h^{2})$$\\!\\!\mathcal{O}(h^{3})$$h$$\lVert
u-u_{h}\rVert_{L^{2}(\Omega)}$$k=15$$p=1$$p=2$$10^{-2}$$10^{-1}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$\\!\\!\mathcal{O}(h^{2})$$\\!\\!\mathcal{O}(h^{3})$$h$$k=20$$p=1$$p=2$
Figure 3: $L^{2}$-error of the approximation of the Helmholtz problem with
homogeneous Dirichlet boundary conditions against a computed reference
solution for $k\in\\{10,12,15,20\\}$. The vertical lines indicates when
$\lambda_{h}^{(i_{\ast})}<k^{2}$ after which we expect quasi-optimality.
$10^{-2}$$10^{-1}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$\\!\\!\mathcal{O}(h^{2})$$\\!\\!\mathcal{O}(h^{3})$$h$$\lVert
u-u_{h}\rVert_{L^{2}(\Omega)}$$k=10$$p=1$$p=2$$10^{-2}$$10^{-1}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$\\!\\!\mathcal{O}(h^{2})$$\\!\\!\mathcal{O}(h^{3})$$h$$k=20$$p=1$$p=2$
Figure 4: $L^{2}$-error of the approximation of the Helmholtz problem with
Robin boundary conditions against a computed reference solution for
$k\in\\{10,20\\}$. The vertical lines indicate when
$\lambda_{h}^{(i_{\ast})}<k^{2}$ after which we expect quasi-optimality.
$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$\delta$$\lVert
u-u_{h}\rVert_{L^{2}(\Omega)}$$\lambda_{i^{*}}=2\pi^{2}$$h=5\times
10^{-3}$$h=2.5\times 10^{-3}$$h=1.25\times
10^{-3}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{2}$$10^{3}$$10^{4}$$10^{5}$$10^{6}$$h$$\lambda_{i^{*}}=2\pi^{2}\approx
19.7392,\;k^{2}\approx 19.739$$p=1$$p=2$ Figure 5: Considering a wave number
$k$ increasing near an eigenvalue $\lambda^{(i_{\ast})}$ the error $\lVert
u-u_{h}\rVert_{L^{2}(\Omega)}$ grows and we loose quasi-optimality. A possible
cure is to increase the polynomial degree $p$ .

###  6.2 The tuning fork

A more challenging geometry is the tuning fork domain. In Figure 8 we show the
results of the numerical experiments for the Helmholtz problem with $k$ equal
to 10 and Dirichlet boundary conditions. In particular, we consider as data
the Gaussian bump

$f(x,y)=5\times
10^{4}\exp\left[{-(40^{2})\cdot((x-10)^{2}+(y-\frac{7}{4})^{2})}\right].$

We solved the Helmholtz problem on a sequence of uniformly refined meshes
until the condition $k^{2}-\lambda^{(i_{\ast})}_{h}$ was positive and then
solved the problem on an additional uniform refinement for comparison. The
value of $k^{2}-\lambda^{(i_{\ast})}_{h}$ for the different meshes is shown in
Table 1. Before the condition $k^{2}-\lambda^{(i_{\ast})}_{h}>0$ was satisfied
we clearly see inspecting Figure (8) that we have unwanted symmetry in the
solution. We also adopted the adaptive$\,{}^{2}$ scheme described in Section
5, and the value of $k^{2}-\lambda^{(i_{\ast})}_{h}$ are shown in Table 2. We
observe that in the case of the adaptive$\,{}^{2}$ scheme, the condition
$k^{2}-\lambda^{(i_{\ast})}_{h}>0$ is once again sufficient to ensure quasi-
optimality and the symmetry in the solution is broken, but we achieve this
result with a much coarser mesh.

###  6.3 The scatterer

We also consider the Helmholtz problem on a scatterer domain. In Figure 9 we
show the results of the numerical experiments for the Helmholtz problem with
$k$ equal to 100 and mixed Dirichlet and Robin boundary conditions. Once
again, we consider as data the Gaussian bump

$f(x,y)=5\times
10^{4}\exp\left[{-(40^{2})\cdot((x-\frac{1}{2})^{2}+(y-\frac{1}{2})^{2})}\right].$

In Table 3 we show the value of
$(k^{2}-\lambda^{(i_{\ast})}_{h})(\lambda^{(i_{\ast})}_{h})^{-1}$ for the
different meshes. From the interference pattern in the solution behind the
scatterer, we can see that when $k^{2}-\lambda^{(i_{\ast})}_{h}$ is
sufficiently small even if negative we have an acceptable solution to the
Helmholtz problem. Such observation is perfectly in line with the theoretical
results presented in Section 4. In fact, we only claim that the condition
$k^{2}-\lambda^{(i_{\ast})}_{h}>0$ is sufficient to ensure quasi-optimality
not necessary. Furthermore, quasi-optimality is not necessary to obtain a good
approximation of the solution to the Helmholtz problem.

(a) $NumberofDoFs:21521$.

(b) $NumberofDoFs:84769$.

(c) $NumberofDoFs:336449$.

(d) $NumberofDoFs:1340545$.

Figure 6: We solve the Helmholtz problem on a tuning fork domain using a sequence of uniformly refined meshes, with $k$ equal to $10$ and data a Gaussian bump $f(x,y)=5\times 10^{4}\exp\left[{-(40^{2})\cdot((x-10)^{2}+(y-\frac{7}{4})^{2})}\right]$. The criteria here proposed suggest that mesh $(c)$ is the first one to guarantee quasi-optimality. Table 1: The table shows $k^{2}-\lambda^{(i_{\ast})}_{h}$ for different degrees of freedom corresponding to meshes obtained by uniform refinement. N. DoFs | 409 | 1469 | 5545 | 21521 | 84769 | 336449
---|---|---|---|---|---|---
$k^{2}-\lambda^{(i_{\ast})}$ | -4.82e+01 | -1.71e+01 | -4.22e+00 | -8.30e-01 | -9.15e-03 | 1.93e-01
Table 2: The table shows $k^{2}-\lambda^{(i_{\ast})}_{h}$ for different
degrees of freedom corresponding to meshes obtained by the adaptive$\,{}^{2}$
scheme described in Section 5, with $\ell=7$.

N. DoFs | 409 | 1445 | 4784 | 5353 | 17829
---|---|---|---|---|---
$k^{2}-\lambda^{(i_{\ast})}_{h}$ | -4.82e+01 | -1.72e+01 | -4.79e+00 | -4.29e+00 | -1.06e+00

N. DoFs 23780 72834 77615 86733 161102 279034 $k^{2}-\lambda^{(i_{\ast})}_{h}$
-7.82e-01 -5.64e-02 -3.07e-02 -8.14e-03 8.08e-02 1.73e-01

Figure 7: The figure shows one of the meshes obtained by the
adaptive$\,{}^{2}$ scheme described in section 5.

(a) $NumberofDoFs:77615$.

(b) $NumberofDoFs:86733$.

(c) $NumberofDoFs:161102$.

(d) $NumberofDoFs:279034$.

Figure 8: We solve the Helmholtz problem on a tuning fork domain with a
sequence of meshes obtain via the adaptive$\,{}^{2}$ scheme, with $k$ equal to
$10$ and data a Gaussian bump $f(x,y)=5\times
10^{4}\exp\left[{-(40^{2})\cdot((x-10)^{2}+(y-\frac{7}{4})^{2})}\right]$. The
criteria here proposed suggest that mesh $(c)$ is the first one to guarantee
quasi-optimality.

(a) $NumberofDoFs:6378$.

(b) $NumberofDoFs:25188$.

(c) $NumberofDoFs:100104$.

(d) $NumberofDoFs:399120$.

Figure 9: We solve the Helmholtz problem on scatterer geometry with a sequence of uniformly refined meshes, with $k$ equal to $100$ and data a Gaussian bump $f(x,y)=5\times 10^{4}\exp\left[{-(40^{2})\cdot((x-\frac{1}{2})^{2}+(y-\frac{1}{2})^{2})}\right]$. The criteria here proposed suggest that mesh $(d)$ is the first one to guarantee quasi-optimality. Table 3: The table shows $(k^{2}-\lambda^{(i_{\ast})}_{h})(\lambda^{(i_{\ast})}_{h})^{-1}$ for different degrees of freedom corresponding to meshes obtained by uniform refinement for the scatterer geometry. N. DoFs | 1635 | 6378 | 25188 | 100104 | 399120
---|---|---|---|---|---
$(k^{2}-\lambda^{(i_{\ast})})(\lambda^{(i_{\ast})}_{h})^{-1}$ | 1.89e+05 | 4.36e+02 | 4.52e+01 | 2.76e+00 | 7.89e-01

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