Over the last 10 years, a wavenumber-explicit convergence theory for the -FEM applied to the Helmholtz equation
an analytic component, satisfying bounds with the same -dependence as those satisfied by the full Helmholtz solution, and
a component with finite regularity, satisfying bounds with improved -dependence compared to those satisfied by the full Helmholtz solution.
Such a decomposition was obtained for
the Helmholtz exterior Dirichlet problem where the obstacle has analytic boundary [34, Theorem 4.20],
This decomposition was then used to prove quasioptimality of the -FEM applied to the standard Helmholtz variational formulation in , , , and applied to a discontinuous Galerkin formulation in . Indeed, for the standard variational formulation (defined for the full-space problem in Definition 2.2 below) applied to the boundary value problems above, provided the solution operator of the problem is bounded polynomially in (see Definition 2.6 below), then there exist and (independent of , and ) such that if
then the Galerkin solution exists, is unique, and satisfies
where is the approximation space and the norm is the standard weighted norm (defined by (2.10
) below). Since the total number of degrees of freedom of the approximation space is proportional to, the significance of this result is that it shows there is a choice of and such that the Galerkin solution is quasioptimal, with quasioptimality constant (i.e. ) independent of , and with the total number of degrees of freedom proportional to ; thus, with these choices of and , the -FEM does not suffer from the pollution effect .
Over the last few years, there has been increasing interest in the numerical analysis of the heterogeneous Helmholtz equation, i.e. the Helmholtz equation with variable coefficients
see, e.g., , , , , , , , , . However there do not yet exist in the literature analogous results to those in , , ,  for the variable-coefficient Helmholtz equation.
1.2 Informal statement and discussion of the main results
The main results.
This paper considers the variable-coefficient Helmholtz equation (1.4) with coefficients posed in , with the Sommerfeld radiation condition at infinity. We obtain analogous results to those obtained in  for this scenario with constant coefficients. That is, we prove quasioptimality of the -FEM under the conditions (1.3) and provided that the solution operator is polynomially bounded in ; see Theorem 3.4 below.
We obtain this result by decomposing the solution to (1.4) into two components:
where and is analytic in , where denotes the ball of radius centred at the origin (and is arbitrary); see Theorem 3.1 below. This is exactly analogous to the decomposition obtained in , except that now satisfies the variable-coefficient equation (1.4) instead of (1.1).
As announced in the abstract , the forthcoming paper  obtains analogous results for the variable-coefficient Helmholtz equation with piecewise-analytic coefficients. The present paper and  therefore contain the first results on the wavenumber-explicit convergence of -FEM for the Helmholtz equation with variable coefficients.
Overview of the ideas behind the decomposition and subsequent bounds.
The idea in  was to decompose the data in (1.1) into “low-” and “high-” frequency components, with the Helmholtz solution for the low-frequency component of and the Helmholtz solution for the high-frequency component of . The frequency cut-offs were defining using the indicator function
with a free parameter (see [33, Equation 3.31] and the surrounding text). In  the frequency cut-off (1.5) was then used with (a) the expression for as a convolution of the fundamental solution and the data , and (b) the fact that the fundamental solution is known explicitly when and , to obtain the appropriate bounds on and using explicit calculation.
In this paper we use the same idea as in  of decomposing into low- and high-frequency components, but apply frequency cut-offs to the solution as opposed to the data . Then, given any cut-off function that is zero for , bounding the corresponding low-frequency component
is relatively straightforward using basic properties of the Fourier-transform (namely the expression for the Fourier transform of a derivative and Parseval’s theorem). Indeed, in Fourier space each derivative corresponds to a power of the Fourier variable, and the frequency cut-off means that for ; i.e. every derivative of brings down a power of compared to (see §5.3 below). The main difficulty therefore is showing that the high-frequency component satisfies a bound with one power of improvement over the bound satisfied by .
The main idea of the present paper is that the high-frequency cut-off can be chosen so that the (scaled) Helmholtz operator
is semiclassically elliptic on the support of the high-frequency cut-off. Furthermore, choosing the cut-off function to be smooth (as opposed to discontinuous, as in (1.5)) then allows us to use basic facts about the “nice” behaviour of elliptic semiclassical pseudodifferential operators (namely, they are invertible up to a small error) to prove the required bound on . (Recall that semiclassical pseudodifferential operators are just pseudodifferential operators with a large/small parameter; in this case the large parameter is .)
We now discuss further the frequency cut-offs and the bound on via ellipticity.
The frequency cut-offs.
In constrast to (1.5), we choose such that
With the Fourier transform and its inverse defined by
we define the low-frequency cut-off by
and the high-frequency cut-off by
so that . We let be equal to one on and vanish outside , and then
The bound on the high-frequency component via ellipticity.
Recall that a PDE is elliptic if its principal symbol is non-zero. The concept of ellipticity for semiclassical differential operators (or, more generally, semiclassical pseudodifferential operators) is analogous, except that it now involves the semiclassical principal symbol (see (4.20) below). The semiclassical principal symbol of (1.6) is
where denotes the inner product and (see (4.13) below and the surrounding text).
If the parameter in the cut-off function (1.7) is chosen to be a certain function of and (see (5.7) below), then the symbol (1.12) is bounded away from zero when , i.e. in the region of Fourier space where is non-zero; one therefore describes as “microlocally elliptic”, where the adjective “microlocal” indicates that we have ellipticity on just a region of phase space (rather than on all of phase space in the more familiar global ellipticity). This microlocal-ellipticity observation alone is enough to prove the required bound on when the solution operator has the best possible behaviour in (see §5.4.1), which occurs when the coefficients and are such that the problem is nontrapping. The bound on when the solution operator is polynomially bounded in
follows from a further use of microlocal ellipticity to estimate a commutator term; see §5.4.2, in particular Lemmas 5.3 and 5.6.
These ellipticity properties are then used with the standard microlocal elliptic estimate for pseudodifferential operators, appearing in the semiclassical setting in, e.g., [15, Appendix E], and stated in this setting as Theorem 4.3 below. The whole point is that a semiclassical pseudodifferential operator that is elliptic in some region of phase space can be inverted (up to some small error) in that region, and the norm of the inverse is bounded uniformly in the large parameter (here ) as long as one uses weighted norms (analogous to the familiar norm (2.10)).
The result is that satisfies a bound with one power of improvement over the bound satisfied by (compare (3.1) and (2.12)). To give a simple illustration of how ellipticity can give this improved -dependence, we contrast the solutions of
with both equations posed in with compactly-supported , and with satisfying the Sommerfeld radiation condition (1.2) and satisfying boundedness at infinity. The bounds that are sharp in terms of -dependence are
with the former given by Part (i) of Theorem 2.7, and the latter following from the Lax-Milgram theorem. The operator is not semiclassically elliptic on all of phase space (its semiclassical principal symbol is ), whereas is semiclassically elliptic on all of phase space (its semiclassical principal symbol is ); we therefore see that ellipticity has resulted in the solution operator having improved -dependence. The proof of the bound on is more technical, but the idea – that the improvement in -dependence comes from ellipticity – is the same.
The assumption that the solution operator is polynomially bounded in .
We need to assume that the solution operator is polynomially bounded in (in sense of Definition 2.6 below), both in proving the bound on , and in proving quasi-optimality of the -FEM.
The -dependence of the Helmholtz solution operator depends on whether the problem is trapping or nontrapping. For the heterogeneous Helmholtz equation (1.4) posed in (i.e. with no obstacle), trapping can be created by the coefficients and ; see, e.g., . If the problem is nontrapping, then the Helmholtz solution operator (measured in the natural norms) is bounded in . However, under the strongest form of trapping, the Helmholtz solution operator can grow exponentially in . Nevertheless, it has recently been proved that, if a set of frequencies of arbitrarily small measure is excluded, then the solution operator is polynomially bounded under any type of trapping . Therefore, the result that the -FEM is quasi-optimal holds for a wide class of Helmholtz problems; see Corollary 3.5 below.
Why do we need coefficients?
As highlighted above, our proof of the decomposition relies on standard results about semiclassical pseudodifferential operators (recapped in §4). These results are usually stated for symbols, and thus to fit into this framework and must be . However, examining the results we use, we see that we only need the symbol of the PDE to be in where depends only on the dimension and on the exponent appearing in the assumption that the solution operator is polynomially bounded (see Definitions 2.5 and 2.6 below). Therefore, while we consider to easily use results about semiclassical pseudodifferential operators from , [15, Appendix E], our results will hold for and , where .
Extending the decomposition result to the solution of other PDEs.
Our proof of the decomposition result only relies on the principal symbol of the differential operator being bounded below at infinity (in the sense of (3.8) below). Therefore, the decomposition result Theorem 3.1 is valid for a much larger class of PDEs (and indeed pseudodifferential operators) than (1.4); see Remark 3.7 below for more details.
Outline of the paper.
§2 gives the definitions of the boundary-value problem and the finite-element method. §3 states the main results. §4 recaps results about semiclassical pseudodifferential operators, with  and [15, Appendix E] as the main references. §5 proves the result about the decomposition (Theorem 3.1). §6 proves the result about quasioptimality of the -FEM (Theorem 3.4). Appendix A contains the proof of Lemma 5.4 used in §5.
2 Formulation of the problem
2.1 The boundary value problem
Assumption 2.1 (Assumptions on the coefficients)
(where is the set of real, symmetric, positive-definite matrices) is such that is compact in and there exist such that, in the sense of quadratic forms,
is such that is compact in and there exist such that
Let be such that , where denotes the ball of radius about the origin and denotes compact containment. Let and denote the Dirichlet and Neumann traces, respectively, on
, where the normal vector for the Neumann trace points out of.
Define to be the Dirichlet-to-Neumann map for the equation posed in the exterior of with the Sommerfeld radiation condition (1.2). The definition of in terms of Hankel functions and polar coordinates (when )/spherical polar coordinates (when ) is given in, e.g., [33, Equations 3.7 and 3.10].
Definition 2.2 (Heterogeneous Helmholtz Problem on )
Given and satisfying Assumption 2.1, such that , , and , satisfies the Heterogeneous Helmholtz Problem on if satisfies the variational problem
where denotes the duality pairing on that is linear in the first argument and antilinear in the second.
Lemma 2.3 (Helmholtz boundary value problems included in Definition 2.2)
with , then , where is the solution to
and satisfies the Sommerfeld radiation condition (1.2) (with replaced by ).
where with and , , where is the solution of the Helmholtz plane-wave scattering problem; i.e.
and satisfies the Sommerfeld radiation condition (1.2) (with replaced by ).
The solution of the Heterogeneous Helmholtz Problem on (defined in Definition 2.2) exists, is unique, and there exists such that
Proof. Uniqueness follows from the unique continuation principle; see [21, §1], [22, §2] and the references therein. Since satisfies a Gårding inequality (see (6.4) below), Fredholm theory then gives existence and the bound (2.7).
Properties of and .
We use later the following two properties of : (i) there exist such that
for all and for all , and (ii)
Let the weighted norm, , be defined by
2.2 The behaviour of the solution operator for large
Definition 2.5 ()
How depends on is crucial to the analysis below, and to emphasise this we write . Below we consider with different values of , and we then write, e.g., (as in the bound (3.2) below).
A key assumption in the analysis of the Helmholtz -FEM is that is polynomially bounded in in the following sense.
Definition 2.6 ( is polynomially bounded in )
Given and , is polynomially bounded for if there exists and such that
where and are independent of (but depend on and possibly also on ).
There exist coefficients and such that for with as , see , but this exponential growth is the worst-possible, since for all by [6, Theorem 2]. We now recall results on when is polynomially bounded in .
Theorem 2.7 (Conditions under which is polynomially bounded in )
(i) and are and nontrapping (i.e. all the trajectories of the Hamiltonian flow defined by the symbol of (1.4) starting in leave after a uniform time), then is independent of .
(ii) If and is then, given and there exists a set with such that
for any , where depends on and . If is for some then the exponent is reduced to .
References for the proof.
(i) is proved using either (a) the propagation of singularities results of  combined with either the parametrix argument of [43, Theorem 3]/ [44, Chapter 10, Theorem 2] or Lax–Phillips theory , or (b) the defect-measure argument of [7, Theorem 1.3 and §3]. It has recently been proved that, for this situation, is proportional to the length of the longest trajectory in ; see [17, Theorems 1 and 2, and Equation 6.32].
(ii) is proved in [27, Theorem 1.1 and Corollary 3.6].
2.3 The finite-element method
Let be a sequence of finite-dimensional subspaces of that converge to in the sense that, for all ,
Later we specialise to the triangulations described in [33, §5], which allow curved elements and thus fit exactly.
3 Statement of the main results
Theorem 3.1 (Decomposition of the solution)
If is polynomially bounded (in the sense of Definition 2.6) for , then there exist such that
where and depend on , and , but are independent of , , , and .
Remark 3.2 ( is analytic)
Remark 3.3 (The bounds of Theorem 3.1 written with the notation )
The following result about quasioptimality of the -FEM is then obtained by combining Theorem 3.1, well-known results about the convergence of the Galerkin method based on duality arguments (recapped in Lemma 6.4 below), and results about the approximation spaces in [33, §5] (used in Lemma 6.5 below).
Theorem 3.4 (Quasioptimality of the -FEM if is polynomially bounded)
Let or , and let . Let be the piecewise-polynomial approximation spaces described in [33, §5] (where, in particular, the triangulations are quasi-uniform), and let be the Galerkin solution defined by (2.15).
Corollary 3.5 (Quasioptimality under specific conditions on and )
Let or , and let .
(i) If and are nontrapping, then there exist , depending on , and , and , but independent of , , and , such that if (1.3) holds then, for all , the Galerkin solution exists, is unique, and satisfies the quasi-optimal error bound (3.3) with given by (3.4).
(ii) If is and then, given , there exist a set with and constants , with all three depending on , , and , but independent of , and additionally depending on and such that, for all , if (1.3) holds (with replaced by ) then the Galerkin solution exists, is unique, and satisfies (3.3) with given by (3.4).
For the plane-wave scattering problem (i.e. for given by (2.6)), the regularity result
was recently proved in [26, Theorem 9.1 and Remark 9.10], where depends on and , but is independent of . The polynomial approximation bounds in [33, §B] imply that, for the sequence of approximation spaces described in [33, §5],
where only depends on the constants in [33, Assumption 5.2] (which depend on the element maps from the reference element). Using (3.6) and (3.5) to bound the right-hand side of (3.3), we obtain the following bound on the relative error of the Galerkin solution.
Corollary 3.6 (Bound on the relative error of the Galerkin solution)
Let the assumptions of Theorem 3.4(i) hold and, furthermore, let be given by (2.6) (so that is the solution of the plane-wave scattering problem). Then there exists , independent of , , and , such that if (1.3) holds, then
for all with given by (3.4); i.e. the relative error can be made arbitrarily small by making smaller.
Remark 3.7 (Theorem 3.1 is valid for solutions of a much larger class of PDEs)
Inspecting the proof of Theorem 3.1 below, we see that the conclusion, i.e. the decomposition with and satisfying the bounds (3.1) and (3.2) respectively, holds under much weaker assumptions. Indeed, the conclusion still holds under the following three assumptions only.
(i) is a family of properly-supported second-order pseudo-differential operators, with principal symbol ,
(ii) is coercive at infinity in the sense that
where does not depend on , and
(iii) the solution to , posed in with and , satisfies the bound
with and independent of , , and . (In fact, the in the on the left-hand side of the bound can be replaced by any number .)
In particular, no assumption is made about lower-order terms of , or the behaviour of at infinity (such as a radiation condition).
4 Recap of relevant results about semiclassical pseudodifferential operators
The proof of Theorem 3.1 relies on standard results about semiclassical pseudodifferential operators. We review these here, with our default references being  and [15, Appendix E]. Homogeneous – as opposed to semiclassical – versions of the results in this section can be found in, e.g., [42, Chapter 7], [38, Chapter 7], [24, Chapter 6].***The counterpart of “semiclassical” involving differential/pseudodifferential operators without a small parameter is usually called “homogeneous” (owing to the homogeneity of the principal symbol) rather than “classical.” “Classical” describes the behaviour in either calculus in the small- or high-frequency limit respectively, where commutators of operators become Poisson brackets of symbols, hence classical particle dynamics replaces wave motion.
While the use of homogeneous pseudodifferential operators in numerical analysis is well established, see, e.g., , , there has been less use of semiclassical pseudodifferential operators. However, these are ideally-suited for studying the high-frequency behaviour of Helmholtz solutions. Indeed, semiclassical pseudodifferential operators are just pseudodifferential operators with a large/small parameter, and behaviour with respect to this parameter is then explicitly kept track of in the associated calculus.
The semiclassical parameter .
Instead of working with the parameter and being interested in the large- limit, the semiclassical literature usually works with a parameter and is interested in the small- limit. So that we can easily recall results from this literature, we also work with the small parameter , but to avoid a notational clash with the meshwidth of the FEM, we let (the notation comes from the fact that the semiclassical parameter is related to Planck’s constant, which is written as ; see, e.g., [47, §1.2], [15, Page 82], [30, Chapter 1]). In this notation, the Helmholtz equation becomes
While some results in semiclassical analysis are valid in the limit small, the results we recap in this section are valid for all with arbitrary.
The semiclassical Fourier transform .
The semiclassical Fourier transform is defined for by
and its inverse by
see [47, §3.3]. Then
Semiclassical Sobolev spaces.
In the same way that it is convenient to work with the weighted norm (2.10) when studying the Helmholtz equation with parameter , it is convenient to use norms weighted with when studying (4.1). Therefore on the space
we use the norm
The set of all possible positions and momenta (i.e. Fourier variables) is denoted by ; this is known informally as “phase space”. Strictly, , but for our purposes, we can consider as .
Symbols, quantisation, and semiclassical pseudodifferential operators.
A symbol is a function on that is also allowed to depend on , and thus can be considered as an -dependent family of functions. Such a family , with , is a symbol of order , written as , if for any multiindices