A hybrid two-level weighted Schwartz method for time-harmonic Maxwell equations

TL;DR

This paper introduces a novel two-level hybrid Schwarz preconditioner for time-harmonic Maxwell equations, demonstrating mesh-independent convergence and efficiency through theoretical analysis and numerical experiments.

Contribution

It develops a new hybrid two-level Schwarz preconditioner with a specialized coarse space for Maxwell equations, ensuring robustness and efficiency.

Findings

Preconditioner achieves mesh-independent GMRES convergence.

Numerical results confirm theoretical robustness and efficiency.

Economical variant avoids solving generalized eigenvalue problems.

Abstract

This paper concerns the preconditioning technique for discrete systems arising from time-harmonic Maxwell equations with absorptions, where the discrete systems are generated by N\'ed\'elec finite element methods of fixed order on meshes with suitable size. This kind of preconditioner is defined as a two-level hybrid form, which falls into the class of ``unsymmetrically weighted'' Schwarz method based on the overlapping domain decomposition with impedance boundary subproblems. The coarse space in this preconditioner is constructed by specific eigenfunctions solving a series of generalized eigenvalue problems in the local discrete Maxwell-harmonic spaces according to a user-defined tolerance $\rho$. We establish a stability result for the considered discrete variational problem. Using this discrete stability, we prove that the two-level hybrid Schwarz preconditioner is robust in the sense that the convergence rate of GMRES is independent of the mesh size, the subdomain size and the wave-number when $\rho$ is chosen appropriately. We also define an economical variant that avoids solving generalized eigenvalue problems. Numerical experiments confirm the theoretical results and illustrate the efficiency of the preconditioners.