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

Linear continuous interior penalty finite element method for Helmholtz equation With High Wave Number: One-Dimensional Analysis

This article addresses the properties of continuous interior penalty (CIP) finite element solutions for the Helmholtz equation. The h-version of the CIP finite element method with piecewise linear approximation is applied to a one-dimensional (1D) model problem. We first show discrete well posedness and convergence results, using the imaginary part of the stabilization operator, for the complex Helmholtz equation. Then we consider a method with real valued penalty parameter and prove an error estimate of the discrete solution in the H1-norm, as the sum of best approximation error plus a pollution term that is the order of the phase difference. It is proved that the pollution effect can be eliminated by selecting the penalty parameter appropriately. As a result of this analysis, thorough and rigorous understanding of the error behavior throughout the range of convergence is gained. Numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior. In particular, we give numerical evidence that the optimal penalty parameter obtained in the 1D case also works very well for the CIP-FEM on two-dimensional Cartesian grids

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