TY - JOUR

T1 - Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation

AU - Gong, Shihua

AU - Gander, Martin J.

AU - Graham, Ivan G.

AU - Lafontaine, David

AU - Spence, Euan A.

N1 - Funding Information:
We gratefully acknowledge support from the UK Engineering and Physical Sciences Research Council Grants EP/R005591/1 (DL and EAS) and EP/S003975/1 (SG, IGG, and EAS). This research made use of the Balena High Performance Computing (HPC) Service at the University of Bath.

PY - 2022/10/31

Y1 - 2022/10/31

N2 - We analyse parallel overlapping Schwarz domain decomposition methods for the Helmholtz equation, where the exchange of information between subdomains is achieved using first-order absorbing (impedance) transmission conditions, together with a partition of unity. We provide a novel analysis of this method at the PDE level (without discretization). First, we formulate the method as a fixed point iteration, and show (in dimensions 1, 2, 3) that it is well-defined in a tensor product of appropriate local function spaces, each with L2 impedance boundary data. We then obtain a bound on the norm of the fixed point operator in terms of the local norms of certain impedance-to-impedance maps arising from local interactions between subdomains. These bounds provide conditions under which (some power of) the fixed point operator is a contraction. In 2-d, for rectangular domains and strip-wise domain decompositions (with each subdomain only overlapping its immediate neighbours), we present two techniques for verifying the assumptions on the impedance-to-impedance maps that ensure power contractivity of the fixed point operator. The first is through semiclassical analysis, which gives rigorous estimates valid as the frequency tends to infinity. At least for a model case with two subdomains, these results verify the required assumptions for sufficiently large overlap. For more realistic domain decompositions, we directly compute the norms of the impedance-to-impedance maps by solving certain canonical (local) eigenvalue problems. We give numerical experiments that illustrate the theory. These also show that the iterative method remains convergent and/or provides a good preconditioner in cases not covered by the theory, including for general domain decompositions, such as those obtained via automatic graph-partitioning software.

AB - We analyse parallel overlapping Schwarz domain decomposition methods for the Helmholtz equation, where the exchange of information between subdomains is achieved using first-order absorbing (impedance) transmission conditions, together with a partition of unity. We provide a novel analysis of this method at the PDE level (without discretization). First, we formulate the method as a fixed point iteration, and show (in dimensions 1, 2, 3) that it is well-defined in a tensor product of appropriate local function spaces, each with L2 impedance boundary data. We then obtain a bound on the norm of the fixed point operator in terms of the local norms of certain impedance-to-impedance maps arising from local interactions between subdomains. These bounds provide conditions under which (some power of) the fixed point operator is a contraction. In 2-d, for rectangular domains and strip-wise domain decompositions (with each subdomain only overlapping its immediate neighbours), we present two techniques for verifying the assumptions on the impedance-to-impedance maps that ensure power contractivity of the fixed point operator. The first is through semiclassical analysis, which gives rigorous estimates valid as the frequency tends to infinity. At least for a model case with two subdomains, these results verify the required assumptions for sufficiently large overlap. For more realistic domain decompositions, we directly compute the norms of the impedance-to-impedance maps by solving certain canonical (local) eigenvalue problems. We give numerical experiments that illustrate the theory. These also show that the iterative method remains convergent and/or provides a good preconditioner in cases not covered by the theory, including for general domain decompositions, such as those obtained via automatic graph-partitioning software.

UR - http://www.scopus.com/inward/record.url?scp=85138321775&partnerID=8YFLogxK

U2 - 10.1007/s00211-022-01318-8

DO - 10.1007/s00211-022-01318-8

M3 - Article

AN - SCOPUS:85138321775

VL - 152

SP - 259

EP - 306

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 2

ER -