This paper analyses the following question: let A_j, j = 1,2, be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations ∇⋅ (A_j∇u_j) + k²n_ju_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 A₂(A₁)^− 1 to converge in a k-independent number of iterations for arbitrarily large k? (In other words, for A₁ to be a good left or right preconditioner for A₂?) 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.

本文分析以下问题：设A _j , j = 1,2, 为异构亥姆霍兹方程∇⋅ ( A _j ∇ u _j ) + k ² n的外狄利克雷问题的有限元离散对应的伽辽金矩阵_j u _j = − f。必须有多小\(\|A_{1} -A_{2}\|_{L^{q}}\)和\(\|{n_{1}} - {n_{2}}\|_{ L ^ {q}} \）是（在以下方面ķ为GMRES -dependence）施加到或者\（（\ mathbf {A} _1）^ { - 1} \ mathbf {A} _2 \）或阿₂（甲₁ ) ^{− 1}收敛于任意大k 的与k无关的迭代次数？（换句话说，对于A ₁来说，A ₁是A _{2 的}一个很好的左或右预处理器？）我们证明了回答这个问题的结果，为它们的锐度提供了理论证据，并给出了支持估计的数值实验。我们解决这个问题的动机来自于计算具有随机系数A和n的亥姆霍兹方程的关注量。这样的计算可能需要解决许多确定性亥姆霍兹问题，每个问题都有不同的A和n，并且上述问题的答案决定了先前计算的 Galerkin 矩阵之一的逆可以在多大程度上用作其他 Galerkin 矩阵的预处理器。