Convergence of overlapping domain decomposition methods with PML transmission conditions applied to nontrapping Helmholtz problems

Abstract

We study overlapping Schwarz methods for the Helmholtz equation posed in any dimension with large, real wavenumber and smooth variable wave speed. The radiation condition is approximated by a Cartesian perfectly-matched layer (PML). The domain-decomposition subdomains are overlapping hyperrectangles with Cartesian PMLs at their boundaries. The overlaps of the subdomains and the widths of the PMLs are all taken to be independent of the wavenumber. For both parallel (i.e., additive) and sequential (i.e., multiplicative) methods, we show that after a specified number of iterations -- depending on the behaviour of the geometric-optic rays -- the error is smooth and smaller than any negative power of the wavenumber. For the parallel method, the specified number of iterations is less than the maximum number of subdomains, counted with their multiplicity, that a geometric-optic ray can intersect. These results, which are illustrated by numerical experiments, are the first wavenumber-explicit results about convergence of overlapping Schwarz methods for the Helmholtz equation, and the first wavenumber-explicit results about convergence of any domain-decomposition method for the Helmholtz equation with a non-trivial scatterer (here a variable wave speed).

Figures (15)

• Figure 1.1: The grid formed by the points $\{y_m^\ell\}$ that is used in the definition of a particular $2$-checkerboard of size $4\times 5$, and the subdomain $\Omega_{{\rm int}, j}$ formed by extending $(y_1^1,y_2^1)\times (y^2_2, y^2_3)$
• Figure 1.2: A sequence of lexicographic orderings of a $3 \times 3$ checkerboard that is exhaustive in the sense of Definition \ref{['def:exhaust']}.
• Figure 2.1: The functions $\{\chi^{>}_j\}$ and $\{\chi_j\}$ for two subdomains
• Figure 2.2: A trajectory (in red) following the word $(1,2,3)$ (here $\Omega_1, \Omega_2, \Omega_3$ are interior subdomains of $\Omega$). The blue hatched shading indicates the domain $\mathrm{supp} \chi^{>}_1 \cap \mathrm{supp}(P^2_{\rm s}-P_{\rm s})$, and the purple hatched shading indicates the domain $\mathrm{supp} \chi^{>}_2 \cap \mathrm{supp}(P^3_{\rm s}-P_{\rm s})$. (The points $(x_j,\xi_j)$, $j=1,2,3$, and sub-trajectories $\gamma_2,\gamma_3$ are used in the precise definition of follow in Definition \ref{['def:follow']} below.)
• Figure 3.1: Examples of elements of $\mathcal{X}^j_0$, $\mathcal{X}^j_1$ and $\mathcal{X}^j_2$ for a 3-d subdomain.

Theorems & Definitions (87)

• Theorem : Informal summary of Theorems \ref{['thm:strip']}-\ref{['thm:sweep']} and \ref{['thm:gen']}
• Theorem 1.1: $c\equiv 1$, strip, parallel method
• Theorem 1.2: $c\equiv 1$, strip, forward-backward sweeping
• Theorem 1.3: $c\equiv 1$, checkerboard, parallel method
• Theorem 1.4: $c\equiv 1$, checkerboard, sequential method
• Remark 1.5: Comparing the number of solves for the parallel and sequential methods on a $N_1\times \dots\times N_1$ checkerboard
• Theorem 1.6: General $c$, arbitrary hyperrectangular subdomains, parallel method
• Remark 1.7: The relationship of PML to the optimal subdomain boundary conditions
• Lemma 2.1: Physical error propagation for the parallel method
• Definition : Informal statement of Definition \ref{['def:follow']} (following a word)