High Performance Parallel Solvers for the time-harmonic Maxwell Equations

TL;DR

This paper evaluates various preconditioners for large-scale numerical solutions of the challenging time-harmonic Maxwell equations, focusing on their performance across different computational and physical parameters.

Contribution

It compares the effectiveness of four different preconditioners, including a new tailored approach, for solving Maxwell equations in a parallel computing environment.

Findings

Hiptmair-Xu and Block Low-Rank preconditioners perform favorably.

Performance varies with mesh size, CPU cores, wavelength, and domain size.

Direct LU factorization is used as a benchmark.

Abstract

We consider the numerical solution of large scale time-harmonic Maxwell equations. To this day, this problem remains difficult, in particular because the equations are neither Hermitian nor semi-definite. Our approach is to compare different strategies for solving this set of equations with preconditioners that are available either in PETSc, MUMPS, or in hypre. Four different preconditioners are considered. The first is the sparse approximate inverse, which is often applied to electromagnetic problems. The second is Restricted Additive Schwarz, a domain decomposition preconditioner. The third is the Hiptmair-Xu preconditioner which is tailored to the positive Maxwell equations, a nearby problem. The final preconditioner is MUMPS's Block Low-Rank method, a compressed block procedure. We also compare the performance of this method to the standard LU factorization technique, which is a direct solver. Performance with respect to the mesh size, the number of CPU cores, the wavelength and the physical size of the domain are considered. This work in progress yields temporary conclusions in favour of the Hiptmair-Xu and the Block Low-Rank preconditioners.