1.1 Context, and informal statement of the main result
When solving the Helmholtz equation with the version of the finite-element method (where accuracy is increased by decreasing the meshwidth while keeping the polynomial degree constant), must decrease faster than to maintain accuracy as increases; this is the so-called “pollution effect” .
An explicit expression for the discrete Green’s function for this problem is available, and Ihlenburg and Babuška used this to prove the following two sets of results:
The -FEM is quasioptimal in the semi-norm, with quasioptimality constant independent of , if is sufficiently small; i.e. there exists , independent of and such that, if , then
where is the appropriate conforming subspace of of piecewise polynomials of degree on meshes of width , and is the Galerkin solution; see [31, Theorem 3], [30, Theorem 4.13], [32, Theorem 3.5] (when this result was proved earlier in [1, Theorem 3.2]). The numerical experiments in [31, Figures 8 and 9] then indicated that, when , the condition “ sufficiently small” for quasi-optimality is necessary.
Under an assumption on the data (discussed below), the relative error in the -FEM can be made arbitrarily small by, when , making sufficiently small and, when (and assuming that the data is sufficiently smooth, see [30, Remark 4.28]), making sufficiently small. More precisely, [31, Equation 3.25], [32, Theorem 3.7], [30, Equation 4.5.15, §4.6.4, and Theorem 4.27] prove that there exists , independent of and (but dependent on ) such that, if is sufficiently small, then
where the weighted norm is defined by (1.17) below. The numerical experiments in [31, Figure 11], and [30, Figure 4.13] then indicated that, when , the condition “ sufficiently small” is necessary for the relative error to be bounded (in agreement with the earlier numerical experiments in  for small ).
The quasi-optimality results in Point 1 above have since been generalised to Helmholtz problems in 2 and 3 dimensions (and improved in the case ). Indeed, the fact that the -FEM with is quasioptimal (with constant independent of ) in the full norm when is sufficiently small was proved for the homogeneous Helmholtz equation on a bounded domain with impedance boundary conditions in [36, Proposition 8.2.7] (in the case of constant coefficients) and [27, Theorem 4.5 and Remark 4.6(ii)] (in the case of variable coefficients), and for scattering problems with variable coefficients in [23, Theorem 3]. The fact that the -FEM for is quasioptimal when is sufficiently small was proved in for a variety of constant coefficient Helmholtz problems in [37, Corollary 5.6], [38, Proof of Theorem 5.8], and [24, Theorem 5.1], and for a variety of problems including variable-coefficient Helmholtz problems in [14, Theorem 2.15]; the condition “ sufficiently small” is indicated to be sharp for quasi-optimality by, e.g., the numerical experiments in [14, §4.4].
In contrast, the relative-error bound (1.2) in Point 2 above has not been obtained for any Helmholtz problem in 2 or 3 dimensions, even though numerical experiments indicate that the condition “ sufficiently small” is necessary and sufficient for the relative error to be controllably small; see, e.g., [19, Left-hand side of Figure 3]. The closest-available result is that, if is sufficiently small, then
for the Helmholtz problem posed in a domain with either impedance boundary conditions on or a perfectly matched layer (PML). Indeed, for the PML problem, (1.3) is proved for in [34, Theorem 4.4 and Remark 4.5(iv)] and [24, Theorem 5.4]. For the impedance problem, (1.3) is proved for in [46, Theorem 6.1], for in [19, Corollary 5.2] (following earlier work by ), and for for the variable-coefficient Helmholtz equation in [40, §2.3] (under a nontrapping condition on and ).
The main results of this paper
The two main results are the following:
Theorem 1.4 proves the relative-error bound (1.2) when for scattering of a plane wave by a nontrapping obstacle and/or a nontrapping inhomogeneous medium (modelled by the PDE with variable and ) in 2 or 3 dimensions (see Definition 1.2 below for the precise definition of the boundary-value problems considered). As highlighted above, the numerical experiments in [5, 31, 30] show that “ sufficiently small” is necessary for the relative error of the -FEM with linear elements to be controllably small (independent of ), and so the result of Theorem 1.4 is the sharp bound to which the title of the paper refers.
As highlighted above, these are the first-ever frequency-explicit relative-error bounds on the Helmholtz -FEM in 2 or 3 dimensions.
These two results are proved for a particular class of Helmholtz problems, namely those corresponding to scattering by a plane wave, and not for the equation with general . We highlight that, for this latter class of problems, it is unreasonable to expect a relative-error bound such as (1.2) to hold, and thus the best one can do is prove bounds for a particular class of realistic data (as we do here). For example, consider the 1-d problem (1.1) with
where has compact support in . The solution to (1.1) is then , which oscillates on a scale of , i.e., a smaller scale than when . The finite-element method with, say, and small (and independent of ) will therefore not resolve this solution, and hence a bound such as (1.2) does not hold. This example is nevertheless consistent with the previous results recalled above since (i) the assumptions on the solution in [31, First equation in §3.4] and [32, Definition 3.2] exclude such data , and (ii) with given by (1.5), and , so that , and the error estimate (1.3) holds in this case because, although the absolute error on left-hand side of (1.3) is large, the right-hand side of (1.3) is larger.
1.2 Formulation of the problem
Assumption (Assumptions on the domain and coefficients)
(i) is a bounded open Lipschitz set such that its open complement is connected.
(ii) (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,
(iii) is such that is compact in and there exist such that
Let the scatterer be defined by . Given such that , where denotes the ball of radius about the origin, let . Let and let . Let
denote the outward-pointing unit normal vector field on bothand . We denote by the corresponding Neumann trace on or and the corresponding conormal-derivative trace. We denote by the Dirichlet trace on or .
[Helmholtz plane-wave scattering problem] Given and with , let Given , , and , as in Assumption 1.2, we say satisfies the Helmholtz plane-wave scattering problem if
and satisfies the Sommerfeld radiation condition
as , uniformly in .
Define to be the Dirichlet-to-Neumann map for the equation posed in the exterior of with the Sommerfeld radiation condition (1.9). When , for some , the definition of in terms of Hankel functions and polar coordinates (when )/spherical polar coordinates (when ) is given in, e.g., [37, Equations 3.7 and 3.10]. Let
When Dirichlet boundary conditions are prescribed in (1.8), let ; when Neumann boundary conditions are prescribed, let .
[Variational formulation of the Helmholtz plane-wave scattering problem] With , , , , , and as above, define as the solution of the variational problem
where denotes the duality pairing on that is linear in the first argument and antilinear in the second. Then , where is the solution of the Helmholtz plane-wave scattering problem of Definition 1.2.
The solution of the Helmholtz plane-wave scattering problem of Definition 1.2 exists and is unique.
Uniqueness follows from the unique continuation principle; see [26, §1], [27, §2] and the references therein. Since satisfies a Gårding inequality (see (7.6) below), Fredholm theory then gives existence.
The finite-element method
Let be a family of triangulations of (in the sense of, e.g., [16, Page 67]) that is shape regular (see, e.g., [8, Definition 4.4.13], [16, Page 128]). When Neumann boundary conditions are prescribed in (1.8), let
when Dirichlet boundary conditions are prescribed we impose the additional condition that elements of are zero on . In both cases we then have , with the dimension of proportional to . Our main results, Theorems 1.4 and 1.4 below require to be at least . For such it is not possible to fit exactly with simplicial elements (i.e. when each element of is a simplex), and fitting with isoparametric elements (see, e.g, [16, Chapter VI]) or curved elements (see, e.g., ) is impractical. Some analysis of non-conforming error is therefore necessary, but since this is very standard (see, e.g., [8, Chapter 10]), we ignore this issue here.
1.3 Definitions of quantities involved in the statement of the main results
Throughout the paper we assume that for some fixed and for some fixed . For simplicity we assume throughout that
Given a bounded open set , we let the weighted norm, be defined by
We now define quantities and that appear in the main results (Theorems 1.4 and 1.4). All of these are dimensionless quantities, independent of , , and , but dependent on one or more of , , (indicated below).
By [37, Lemma 3.3], there exist , such that
for all and for all , and
We assume that , , and are nontrapping in the sense that there exists such that, given , the solution of the boundary value problem (BVP)
and satisfies the Sommerfeld radiation condition (1.9) (with replaced by ), satisfies the bound
observe that the factor on the right-hand side makes dimensionless. (Remark 1.4 discusses the situation where this nontrapping assumption is removed and depends on .) This assumption holds if the obstacle and the coefficients and are nontrapping in the sense that all billiard trajectories (or, more precisely, Melrose–Sjöstrand generalized bicharacteristics [29, Section 24.3]) starting in an exterior neighbourhood of and evolving according to the Hamiltonian flow defined by the symbol of (1.8) escape from that neighbourhood after some uniform time. For this flow to be well-defined, must be , and and must be globally and in a neighbourhood of ; note that the flow may in general be set-valued rather than unique in cases where the boundary is permitted to be infinite-order flat. Assuming the uniqueness of the flow, an explicit expression for in terms of and is then given in [23, Theorems 1 and 2, and Equation 6.32]. However, the bound (1.20) can be established in situations with much less smoothness; indeed, [26, Theorems 2.5, 2.7, and 2.19] establishes (1.20) for a Dirichlet star-shaped obstacle and and satisfying certain monotonicity assumptions. Furthermore, our arguments in the rest of the paper do not need the flow to be well-defined on , they only require that the bound (1.20) holds. We can therefore define nontrapping in this weaker sense, and work with scatterers of much lower smoothness than in standard microlocal-analysis settings.
1.4 Statement of the main results
The first theorem holds for any , but is most relevant in the case . Let be the solution of the Helmholtz plane-wave scattering problem (Definition 1.2). Assume that both Assumption 1.2 and (1.16) hold, is , and and are nontrapping. If and
then the Galerkin solution to the variational problem (1.14) exists, is unique, and satisfies the bound
Let be the solution of the Helmholtz plane-wave scattering problem (Definition 1.2). Assume that both Assumption 1.2 and (1.16) hold, , , is a nontrapping Dirichlet obstacle, is analytic, and the triangulation in the definition of (1.13) satisfies the quasi-uniformity assumption [38, Assumption 5.1]. If
then the Galerkin solution to the variational problem (1.14) exists, is unique, and satisfies the bound
Remark (The ideas behind the proofs)
Theorems 1.4 and 1.4 are proved by adapting the so-called “elliptic-projection” argument, used to prove the bound (1.3) on the solution in terms of the data, to instead prove relative-error bounds. The elliptic-projection argument was introduced in the Helmholtz context in [20, 21] for interior-penalty discontinuous Galerkin methods, used for the standard FEM and continuous interior-penalty methods in [46, 48], subsequently used by [4, 47, 13, 24, 34], and then augmented with an “error splitting” argument in  (see, e.g., the literature review in [40, §2.3]). The elliptic-projection argument itself is a modification of the classic “duality argument”, coming out of ideas introduced in , which was used to prove quasi-optimality of the Helmholtz FEM in [1, 31, 36, 41, 37, 38, 13, 14, 24, 27, 23].
Our modifications of the elliptic-projection argument are outlined in §2. Aside from keeping track of how all the constants in this argument depend on and , our two new ingredients are (i) a rigorous proof, using microlocal/semiclassical analysis, of the bound (1.21) describing the oscillatory behaviour of the solution of the plane-wave scattering problem (see Theorem 6 below), and (ii) the proof of regularity, with constant independent of , of the solution of Poisson’s equation with the boundary condition ; see (1.24) and Theorem 3.
Regarding (i): oscillatory behaviour similar to (1.21) of Helmholtz solutions has been an assumption in many analyses of finite- and boundary-element methods; see, e.g., [31, First equation in §3.4], [32, Definition 3.2], [9, Definition 4.6], [3, Definition 3.5], [18, Assumption 3.4]. However, to our knowledge, the only existing rigorous proofs of such behaviour are [25, Theorems 1.1 and 1.2] and [22, Theorem 1.11(c)], both concerning the Neumann trace of the solution of the Helmholtz plane-wave scattering problem with and .
Regarding (ii): the analogous result ( regularity with constant independent of ) for Poisson’s equation with the impedance boundary condition is central to the elliptic-projection argument for the Helmholtz equation with impedance boundary conditions. This result was explicitly assumed in [21, Lemma 4.3], implicitly assumed in [46, 48, 4, 13], and recently proved in .
Remark (Why does Theorem 1.4 not cover scattering by an inhomogeneous medium?)
In both the elliptic-projection argument and the standard duality argument, a key role is played by the quantity defined by (5.3) below, which describes how well solutions of the (adjoint of the) Helmholtz equation can be approximated in .
In the case we estimate using regularity of the solution (which holds when and satisfy the assumptions of Theorem 1.4), leading to the bound (5.5) below. When , , , is a Dirichlet obstacle, and is analytic,  proved the bound (5.6) on , and we use this result to prove Theorem 1.4. The bound (5.6) was proved via a judicious splitting of the solution [38, Theorem 4.20] into an analytic but oscillating part, and an part that behaves “well” for large frequencies, and this splitting is only available for the exterior Dirichlet problem with and .
We highlight that an alternative splitting procedure valid for Helmholtz problems with variable coefficients was recently developed in , leading to an alternative proof of the bound on (5.6) [14, Lemma 2.13]. However, this alternative procedure requires that be approximated by on . Indeed, in [14, Proof of Lemma 2.13] the solution is expanded in powers of , i.e. , and then on one has ; this relationship between and on no longer holds if is not approximated by .
Remark (Approximating )
Implementing the operator is computationally expensive, and so in practice one seeks to approximate this operator by either imposing an absorbing boundary condition on , or using a PML. In this paper we follow the precedent established in [37, 38] of, when proving new results about the FEM for exterior Helmholtz problems, first assuming that is realised exactly. We remark, however, that if the two key ingredients in Remark 1.4 (a proof of the oscillatory behaviour (1.21) and -regularity, independent of , of a Poisson problem) can be established when is replaced by an absorbing boundary condition on , then the result of Theorem 1.4 carry over to this case. When an impedance boundary condition (i.e. the simplest absorbing boundary condition) is imposed on , the necessary Poisson -regularity result is proved in , but we discuss below in Remark 6 the difficulties in proving (1.21) in this case.
Remark (Removing the nontrapping assumption)
The only place in the proofs of Theorems 1.4 and 1.4 where the nontrapping assumption (i.e. the fact that in (1.20) is independent of ) is used is in the proof of the bound (1.21) (in Theorem 6 below). We briefly discuss in Remark 6 below the prospects of proving (1.21) in the trapping case (i.e. when is not independent of ). Once (1.21) is proved in the trapping case, the rest of the proofs of Theorems 1.4 and 1.4 go through as before. In the case of Theorem 1.4, the requirement for the relative error to be bounded independently of would then be that be sufficiently small. Under the strongest form of trapping, can grow exponentially through a sequence of s [7, §2.5], but is bounded polynomially in if a set of frequencies of arbitrarily-small measure is excluded [33, Theorem 1.1]. However, it is not clear how sharp the requirement “ sufficiently small” for the relative error to be bounded is in these cases.
2 Outline of the proof and connection to existing arguments
As in the standard duality argument coming out of ideas introduced in  and then formalised in , our starting point is the fact that, since satisfies the Gårding inequality (7.6), Galerkin orthogonality (1.15) and continuity of (7.4) imply that, for any ,
Recall (e.g. from [37, Eq. 4.9]) that the standard duality argument shows that
where , defined by (5.3) below, describes how well solutions of the adjoint problem are approximated in the space (see [41, Theorems 2.2 and 2.5], [37, Theorems 4.2 and 4.3], [44, Theorem 6.32]). Inputting (2.2) into (2.1) one obtains quasioptimality, with constant independent of , if is sufficiently small, and the bounds on described in Lemma 5 below then imply that this condition is satisfied if is sufficiently small.
In contrast, the elliptic-projection argument, which we follow initially, shows that, if is sufficiently small then
(see Lemma 7 below), where in this overview discussion we use the notation when with independent of and , but dependent on and . Observe that (2.3) is a stronger bound than (2.2), since on the right-hand side of (2.3) is arbitrary. The proof of (2.3) in our setting of the plane-wave scattering problem requires the new Poisson -regularity bound (1.24), which we prove in Theorem 3 below.
on the first term on the right-hand side of (2.1), we obtain that, if is sufficiently small, then, for any ,
Assuming regularity of the solution, and using (1.26), we obtain that, if is sufficiently small, then
In the standard elliptic-projection argument (see, e.g., [13, §5.5]) applied to the PDE , an -regularity bound similar to (1.20) and the nontrapping bound (1.20) are combined to give , and combining this with both (2.5) and the bound (see (5.5) below) proves the bound (1.3) with on the Galerkin error in terms of the data when is sufficiently small.
3 Proof of the Poisson -regularity result (1.24)
Let be a bounded, convex, open set of with boundary. Then, for all ,
where is the surface gradient on and is the tangential component of .
The result with real follows from [28, Theorem 126.96.36.199] and the fact that the second fundamental form of (defined in, e.g., [28, §3.1.1]), is non-positive (see [28, Proof of Theorem 188.8.131.52]). The result with complex follows in a straightforward way by repeating the argument in [28, Theorem 184.108.40.206] for complex . ([28, Lemma 220.127.116.11].) If satisfies (1.6) (with replaced by ), then, for all ,
As a first step to proving Theorem 3, we prove it in the case when .
Let satisfy (1.6) (with replaced by ) and be such that . Given , let be the solution of
Let be the outgoing solution of the following transmission problem