Domain decomposition preconditioners for high-order discretizations of the heterogeneous Helmholtz equation

Abstract

We consider one-level additive Schwarz domain decomposition preconditioners for the Helmholtz equation with variable coefficients (modelling wave propagation in heterogeneous media), subject to boundary conditions that include wave scattering problems. Absorption is included as a parameter in the problem. This problem is discretized using $H^1$-conforming nodal finite elements of fixed local degree $p$ on meshes with diameter $h = h(k)$, chosen so that the error remains bounded with increasing $k$. The action of the one-level preconditioner consists of the parallel solution of problems on subdomains (which can be of general geometry), each equipped with an impedance boundary condition. We prove rigorous estimates on the norm and field of values of the left- or right-preconditioned matrix that show explicitly how the absorption, the heterogeneity in the coefficients and the dependence on the degree enter the estimates. These estimates prove rigorously that, with enough absorption and for $k$ large enough, GMRES is guaranteed to converge in a number of iterations that is independent of $k,p$ and the coefficients. The theoretical threshold for $k$ to be large enough depends on $p$ and on the local variation of coefficients in subdomains (and not globally). Extensive numerical experiments are given for both the absorptive and the propagative cases; in the latter case, we investigate examples both when the coefficients are nontrapping and when they are trapping. These experiments support (i) our theory in terms of dependence on polynomial degree and the coefficients; and (ii) the sharpness of our field of values estimates in terms of the level of absorption required.

A comparison of coarse spaces for Helmholtz problems in the high frequency regime

“…When the problems are large, however,-the case when one discretises the Helmholtz equation accurately for high wave numbers-domain decomposition methods are a natural choice [3]. Nevertheless, despite recent efforts and in view of the latest results obtained both at the theoretical [4,5,6] or numerical level [7,8,9], there is no established method outperforming all others in the case of the Helmholtz problem.…”

Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices

“…In our work, we would like to explore this idea of weak scalability at the continuous level (independent of the discretization) for a strip-wise decomposition into subdomains as it arises naturally in the solution of wave-guide problems. While in [24,26] the family of problems is parameterized by the wave number k and the focus is on k-robustness, here we focus on the weak scalability aspect for a family consisting of a growing chain of fixed-size subdomains. Nonetheless, we will see that k-robustness in certain scenarios can easily be derived from our theory.…”

Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

“…D is the norm on the finite-element space induced by the weighted 1 norm 1 in which the PDE analysis of the Helmholtz equation naturally takes place. (Results about convergence of domaindecomposition methods in the norms (2.1) were recently obtained for the Helmholtz equation in [31,34,35] and for the time-harmonic Maxwell equations in [7]). We further define op ess sup x x 2 , where here 2 denotes the spectral/operator norm on matrices, induced by the Euclidean norm 2 on vectors.…”