Numerical assessment of PML transmission conditions in a domain decomposition method for the Helmholtz equation
Niall Bootland, Sahar Borzooei, Victorita Dolean, Pierre-Henri Tournier
TL;DR
Solving large-scale Helmholtz equations yields ill-conditioned linear systems; the paper introduces a PML-based transmission strategy within an ORAS overlapping domain decomposition preconditioner to enhance convergence in both 2D and 3D. The approach integrates PML interfaces into local subdomain solves, using the preconditioner $M_{\text{ORAS}}^{-1}= \sum_{s=1}^{N_{\text{sub}}} R_s^T D_s A_s^{-1} R_s$ for the global system $A \mathbf{u} = \mathbf{b}$ solved by GMRES with right preconditioning. Key findings show that PML interfaces reduce iteration counts and improve $L^2$ accuracy compared to impedance, with $\sigma_{-1}$ typically offering the best performance across frequencies and discretizations. This work provides a scalable, parallel solver framework for Helmholtz problems, enabling more efficient wave propagation simulations in acoustics and geophysics where large 2D/3D domains are common.
Abstract
The convergence rate of domain decomposition methods (DDMs) strongly depends on the transmission condition at the interfaces between subdomains. Thus, an important aspect in improving the efficiency of such solvers is careful design of appropriate transmission conditions. In this work, we will develop an efficient solver for Helmholtz equations based on perfectly matched layers (PMLs) as transmission conditions at the interfaces within an optimised restricted additive Schwarz (ORAS) domain decomposition preconditioner, in both two and three dimensional domains. We perform a series of numerical simulations on a model problem and will assess the convergence rate and accuracy of our solutions compared to the situation where impedance boundary conditions are used.