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

In $d$ dimensions, approximating an arbitrary function oscillating with frequency $\lesssim k$ requires $\sim k^d$ degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber $k$) suffers from the pollution effect if, as $k\to \infty$, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold. While the $h$-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth $h$ and keeping the polynomial degree $p$ fixed) suffers from the pollution effect, the celebrated papers [Melenk, Sauter 2010], [Melenk, Sauter 2011], [Esterhazy, Melenk 2012], and [Melenk, Parsania, Sauter 2013] showed that the $hp$-FEM (where accuracy is increased by decreasing the meshwidth $h$ and increasing the polynomial degree $p$) applied to a variety of constant-coefficient Helmholtz problems does not suffer from the pollution effect. The heart of the proofs of these results is a PDE result splitting the solution of the Helmholtz equation into "high" and "low" frequency components. In this expository paper we prove this splitting for the constant-coefficient Helmholtz equation in full space (i.e., in $\mathbb{R}^d$) using only integration by parts and elementary properties of the Fourier transform; this is in contrast to the proof for this set-up in [Melenk, Sauter 2010] which uses somewhat-involved bounds on Bessel and Hankel functions. The proof in this paper is motivated by the recent proof in [Lafontaine, Spence, Wunsch 2022] of this splitting for the variable-coefficient Helmholtz equation in full space; indeed, the proof in [Lafontaine, Spence, Wunsch 2022] uses more-sophisticated tools that reduce to the elementary ones above for constant coefficients

Can coercive formulations lead to fast and accurate solution of the Helmholtz equation?

A new, coercive formulation of the Helmholtz equation was introduced in [Moiola, Spence, SIAM Rev. 2014]. In this paper we investigate $h$-version Galerkin discretisations of this formulation, and the iterative solution of the resulting linear systems. We find that the coercive formulation behaves similarly to the standard formulation in terms of the pollution effect (i.e. to maintain accuracy as $k\to\infty$, $h$ must decrease with $k$ at the same rate as for the standard formulation). We prove $k$-explicit bounds on the number of GMRES iterations required to solve the linear system of the new formulation when it is preconditioned with a prescribed symmetric positive-definite matrix. Even though the number of iterations grows with $k$, these are the first such rigorous bounds on the number of GMRES iterations for a preconditioned formulation of the Helmholtz equation, where the preconditioner is a symmetric positive-definite matrix.Comment: 27 pages, 7 figure

Domain Decomposition preconditioning for high-frequency Helmholtz problems with absorption

In this paper we give new results on domain decomposition preconditioners for GMRES when computing piecewise-linear finite-element approximations of the Helmholtz equation $-\Delta u - (k^2+ {\rm i} \varepsilon)u = f$, with absorption parameter $\varepsilon \in \mathbb{R}$. Multigrid approximations of this equation with $\varepsilon \not= 0$ are commonly used as preconditioners for the pure Helmholtz case ($\varepsilon = 0$). However a rigorous theory for such (so-called "shifted Laplace") preconditioners, either for the pure Helmholtz equation, or even the absorptive equation ($\varepsilon \not=0$), is still missing. We present a new theory for the absorptive equation that provides rates of convergence for (left- or right-) preconditioned GMRES, via estimates of the norm and field of values of the preconditioned matrix. This theory uses a $k$- and $\varepsilon$-explicit coercivity result for the underlying sesquilinear form and shows, for example, that if $|\varepsilon|\sim k^2$, then classical overlapping additive Schwarz will perform optimally for the absorptive problem, provided the subdomain and coarse mesh diameters are carefully chosen. Extensive numerical experiments are given that support the theoretical results. The theory for the absorptive case gives insight into how its domain decomposition approximations perform as preconditioners for the pure Helmholtz case $\varepsilon = 0$. At the end of the paper we propose a (scalable) multilevel preconditioner for the pure Helmholtz problem that has an empirical computation time complexity of about $\mathcal{O}(n^{4/3})$ for solving finite element systems of size $n=\mathcal{O}(k^3)$, where we have chosen the mesh diameter $h \sim k^{-3/2}$ to avoid the pollution effect. Experiments on problems with $h\sim k^{-1}$, i.e. a fixed number of grid points per wavelength, are also given

Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

This paper analyses the following question: let $\mathbf{A}_j$, $j=1,2,$ be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations $\nabla\cdot (A_j \nabla u_j) + k^2 n_j u_j= -f$. How small must $\|A_1 -A_2\|_{L^q}$ and $\|{n_1} - {n_2}\|_{L^q}$ be (in terms of $k$-dependence) for GMRES applied to either $(\mathbf{A}_1)^{-1}\mathbf{A}_2$ or $\mathbf{A}_2(\mathbf{A}_1)^{-1}$ to converge in a $k$-independent number of iterations for arbitrarily large $k$? (In other words, for $\mathbf{A}_1$ to be a good left- or right-preconditioner for $\mathbf{A}_2$?). We prove results answering this question, give theoretical evidence for their sharpness, and give numerical experiments supporting the estimates. Our motivation for tackling this question comes from calculating quantities of interest for the Helmholtz equation with random coefficients $A$ and $n$. Such a calculation may require the solution of many deterministic Helmholtz problems, each with different $A$ and $n$, and the answer to the question above dictates to what extent a previously-calculated inverse of one of the Galerkin matrices can be used as a preconditioner for other Galerkin matrices

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

We consider the scalar Helmholtz equation with variable, discontinuous coefficients, modelling transmission of acoustic waves through an anisotropic penetrable obstacle. We first prove a well-posedness result and a frequency-explicit bound on the solution operator, with both valid for sufficiently-large frequency and for a class of coefficients that satisfy certain monotonicity conditions in one spatial direction, and are only assumed to be bounded (i.e., $L^\infty$) in the other spatial directions. This class of coefficients therefore includes coefficients modelling transmission by penetrable obstacles with a (potentially large) number of layers (in 2-d) or fibres (in 3-d). Importantly, the frequency-explicit bound holds uniformly for all coefficients in this class; this uniformity allows us to consider highly-oscillatory coefficients and study the limiting behaviour when the period of oscillations goes to zero. In particular, we bound the $H^1$ error committed by the first-order bulk correction to the homogenized transmission problem, with this bound explicit in both the period of oscillations of the coefficients and the frequency of the Helmholtz equation; to our knowledge, this is the first homogenization result for the Helmholtz equation that is explicit in these two quantities and valid without the assumption that the frequency is small

Wavenumber-explicit convergence of the hphp-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients

A convergence theory for the $hp$-FEM applied to a variety of constant-coefficient Helmholtz problems was pioneered in the papers [Melenk-Sauter, 2010], [Melenk-Sauter, 2011], [Esterhazy-Melenk, 2012], [Melenk-Parsania-Sauter, 2013]. This theory shows that, if the solution operator is bounded polynomially in the wavenumber $k$, then the Galerkin method is quasioptimal provided that $hk/p \leq C_1$ and $p\geq C_2 \log k$, where $C_1$ is sufficiently small, and $C_2$ is sufficiently large. This paper proves the analogous quasioptimality result for the heterogeneous (i.e. variable coefficient) Helmholtz equation, posed in $\mathbb{R}^d$, $d=2,3$, with the Sommerfeld radiation condition at infinity, and $C^\infty$ coefficients. We also prove a bound on the relative error of the Galerkin solution in the particular case of the plane-wave scattering problem