Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption

Abstract

Abstract. In this paper we give new results on domain decomposition preconditioners for GM-RES when computing piecewise-linear finite-element approximations of the Helmholtz equation −∆u − (k 2 + iε)u = f , with absorption parameter ε ∈ R. Multigrid approximations of this equation with ε = 0 are commonly used as preconditioners for the pure Helmholtz case (ε = 0). However a rigorous theory for such (so-called "shifted Laplace") preconditioners, either for the pure Helmholtz equation, or even the absorptive equation (ε = 0), is still missing. We present a new theory for the absorptive equation that provides rates of convergence for (left-or right-) preconditioned GMRES, via estimates of the norm and field of values of the preconditioned matrix. This theory uses a k-and ε-explicit coercivity result for the underlying sesquilinear form and shows, for example, that if |ε| ∼ k 2 , then classical overlapping additive Schwarz will perform optimally for the absorptive problem, provided the subdomain and coarse mesh diameters are carefully chosen. Extensive numerical experiments are given that support the theoretical results. While the theory applies to a certain weighted variant of GMRES, the experiments for both weighted and classical GMRES give comparable results. The theory for the absorptive case gives insight into how its domain decomposition approximations perform as preconditioners for the pure Helmholtz case ε = 0. At the end of the paper we propose a (scalable) multilevel preconditioner for the pure Helmholtz problem that has an empirical computation time complexity of about O(n 4/3 ) for solving finite element systems of size n = O(k 3 ), where we have chosen the mesh diameter h ∼ k −3/2 to avoid the pollution effect. Experiments on problems with h ∼ k −1 , i.e. a fixed number of grid points per wavelength, are also given.

Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?

“…We see that the preconditioner works much better when the discretization is finer, which is natural since the theory in the rest of the paper is based on continuous (as opposed to discrete) arguments. Comparing Table 9 with Tables 6, 7 and 8, we see that the behaviour of d and n GMRES for the trapping domain is quite different to the behaviour of d and n GMRES Table 10 The values of d (and in parentheses n GMRES ) for Example 5.4 with n ∼ k and a different range of k k ε = k/4 ε = k/2 ε = k ε = 2k ε = 4k ε = k 3/2 ε = k 2 5π/2 0.1610 (15) 0.0798 (21) 0.0246 ( for the square. Indeed, whereas 0 is never in the numerical range for the square, it is for the trapping domain for the larger values of ε.…”

“…Indeed, the question of how one should choose ε for (P2) to hold when B −1 ε is constructed using multigrid is considered in the recent preprint [7]. Furthermore, in a subsequent paper [21] we will describe for a class of domain decomposition preconditioners how ε should be chosen for these so that (P2) holds.…”

Stability and finite element error analysis for the Helmholtz equation with variable coefficients

“…The stability theory of the underlying PDE turned out to be key to rigorously understanding the performance of iterative methods in the homogeneous case (e.g. [19], [26]), and so analogous results for the heterogeneous case will again be important in the construction and analysis of efficient solvers.…”

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

“…There, the authors prove that if the Galerkin matrix of the standard formulation of the IIP is preconditioned with the Galerkin matrix of the corresponding problem with absorption added in the form ∆ + k 2 → ∆ + k 2 + iε, then GMRES converges in a k-independent number of iterations when ε/k is sufficiently small. However, finding cheap approximations of the Galerkin matrix under this level of absorption is difficult; see [45]. Therefore, the MS formulation is currently the only formulation in the literature that has both a symmetric, positive-definite preconditioner and a rigorous bound on the number of GMRES iterations when applying the preconditioner (albeit with the number of iterations growing with k).…”