Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method

Abstract

Over the last ten years, results from [51], [52], [25], and [50] decomposing high-frequency Helmholtz solutions into "low"-and "high"-frequency components have had a large impact in the numerical analysis of the Helmholtz equation. These results have been proved for the constant-coefficient Helmholtz equation in either the exterior of a Dirichlet obstacle or an interior domain with an impedance boundary condition.Using the Helffer-Sjöstrand functional calculus [35], this paper proves analogous decompositions for scattering problems fitting into the black-box scattering framework of Sjöstrand-Zworski [66], thus covering Helmholtz problems with variable coefficients, impenetrable obstacles, and penetrable obstacles all at once.In particular, these results allow us to prove new frequency-explicit convergence results for (i) the hp-finite-element method applied to the variable coefficient Helmholtz equation in the exterior of a Dirichlet obstacle, when the obstacle and coefficients are analytic, and (ii) the h-finite-element method applied to the Helmholtz penetrable-obstacle transmission problem.

Wavenumber-Explicit hp-FEM Analysis for Maxwell’s Equations with Transparent Boundary Conditions

“…This technique, often called "Schatz argument", crucially hinges on the regularity of the dual problem, which is again a Helmholtz problem. The key new insight of the line of work [15,33,34] is a refined wavenumber-explicit regularity theory for Helmholtz problems that takes the following form ("regularity by decomposition"): given data, the solution u is written as u H 2 + u A where u H 2 has the regularity expected of elliptic problems and is controlled in terms of the data with constants independent of k. The part u A is a (piecewise) analytic function whose regularity is described explicitly in terms of k. Employing "regularity by decomposition" for the analysis of discretizations has been successfully applied to other Helmholtz problems and discretizations such DG methods [29], BEM [22], FEM-BEM coupling [24], and heterogeneous Helmholtz problems [5,9,20,21]. In this paper, we consider the following time-harmonic Maxwell equations with impedance boundary conditions as our model problem:…”

A simple proof that the $hp$-FEM does not suffer from the pollution effect for the constant-coefficient full-space Helmholtz equation

“…Sketch proof. Use the Sobolev embedding theorem (see, e.g., [38,Theorem 3.26]) to obtain a bound on ∂ α u L ∞ (D) , and then use this to bound the Lagrange form of the remainder in the Taylor series; see, e.g., [35,Appendix C].…”

“…discontinuous Galerkin methods [19,20,40,64] and multiscale methods [26,10,5,54,53,8,12]. Moreover, there is large current interest in this question when the Helmholtz equation (1.1) is replaced by its variable-coefficient generalisation ∇ • (A∇u) + k 2 nu = 0 [7,11,24,28,35,37,34] or even the time-harmonic Maxwell equations [50,43,44]. A highlight of this body of research is the result from [41], [42], [17], and [40] that the hp-FEM does not suffer from the pollution effect ; i.e., accuracy (in the sense that the computed solution is quasi-optimal -see (3.2) below) can be maintained with a choice of the number of degrees of freedom growing like k d .…”

“…In this case the Fourier transform (6.1) is no longer available; nevertheless the functional calculus (essentially the idea of expanding in eigenfunctions of the differential operator) can be used to define Fourier-type transforms, tailored to the particular problem. The ideas of the proof of Theorem 5.1 can then be implemented in this situation (albeit with more technicalities, and the requirement that the obstacle is analytic); see [35].…”

Scattering by finely-layered obstacles: frequency-explicit bounds and homogenization

“…Regarding (ii): the first such bounds were proved for the exterior Helmholtz equation in [8] (for A variable and n = 1) and [9], (for n variable and A = I). Recent such bounds were proved in [59,38,32,79] (and for problems posed on bounded domains with impedance boundary conditions [23,Chapter 2], [6,11,67,78,38,39]), the renewed interest due to growing interest in the numerical analysis of Helmholtz equation with variable coefficients [24,6,11,67,30,33,25,26,70,39,32,71,50,36,51].…”