Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption

Abstract

This paper rigorously analyses preconditioners for the time-harmonic Maxwell equations with absorption, where the PDE is discretised using curl-conforming finite-element methods of fixed, arbitrary order and the preconditioner is constructed using Additive Schwarz domain decomposition methods. The theory developed here shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters and overlap are chosen appropriately, then the classical two-level overlapping Additive Schwarz preconditioner (with PEC boundary conditions on the subdomains) performs optimally -in the sense that GMRES converges in a wavenumberindependent number of iterations -for the problem with absorption. An important feature of the theory is that it allows the coarse space to be built from low-order elements even if the PDE is discretised using high-order elements. It also shows that additive methods with minimal overlap can be robust. Numerical experiments are given that illustrate the theory and its dependence on various parameters. These experiments motivate some extensions of the preconditioners which have better robustness for problems with less absorption, including the propagative case. At the end of the paper we illustrate the performance of these on two substantial applications; the first (a problem with absorption arising from medical imaging) shows the empirical robustness of the preconditioner against heterogeneity, and the second (scattering by a COBRA cavity) shows good scalability of the preconditioner with up to 3,000 processors.

A Two-Level Domain-Decomposition Preconditioner for the Time-Harmonic Maxwell’s Equations

“…Our aim is to extend these ideas to the time-harmonic Maxwell's equations, both from the theoretical and numerical points of view. These results will appear in full in the forthcoming paper Bonazzoli et al [2017].…”

“…We will consider both these cases, in each case constructing preconditioners by using larger values of κ. Indeed, a higher level of absorption makes the problems involved in the preconditioner definition more "elliptic" (in a sense more precisely explained in Bonazzoli et al [2017]), thus easier to solve. Note that the absorption cannot increase too much, otherwise the problem in the preconditioner is "too far away" from the initial problem.…”

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

“…Figure 7 shows, for both 1-and 2-d, the growth of the number of GMRES iterations with k for the situations described in Theorems 4.6 and 4.7 (i.e. left preconditioning) except that Theorems 4.6 and 4.7 are for GMRES with weight D ; we find the behaviour of weighted GMRES essentially identical (note that this situation of the behaviour of GMRES being almost identical in the weighted and unweighted settings was also encountered in the domain-decomposition methods of [44,45]). We also find essentially identical results for right preconditioning.…”

“…In this section, we explore the implications of the refined version of the Elman estimate (4.2) for weighted GMRES, where the weight matrix corresponds to the mass matrix of one of the norms · V1 and · V2 . This weighted setting was inspired by the recent work on domain-decomposition preconditioners for the Helmholtz and Maxwell equations in [45] and [44].…”

“…2 is proved in[44, Corollary 5.4], and implies that choosing m −1 is sufficient for the decrease of the residual to be independent of as → 0.…”