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.

翻译：本文分析如下问题 : 让 $\ mathbf{A\\ j} A\ j= 美元, $j= 1, 2美元, 美元是 Galerkin 矩阵, 相当于 异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异异等( A_ j\ j_ j) + kbla u_ j_ j= - f$ 。 以 任意大异异异异种为异异异类以1美元计数 = - { n1} - 以 美元计数 美元=2\ liqq} 以( 以 美元计算 $- 美元为基异异异异异异异异异异异异异异异异异异异的G 。