Several ways to achieve robustness when solving wave propagation problems

TL;DR

This paper reviews the challenges in solving wave propagation problems, especially at high frequencies and in heterogeneous media, and introduces a new spectral coarse space preconditioner for Maxwell's equations.

Contribution

It provides a comprehensive overview of difficulties in wave problems and proposes a novel subspace decomposition-based preconditioner for heterogeneous media.

Findings

Identifies key difficulties in discretising wave problems.

Demonstrates limitations of current methods on realistic examples.

Introduces a new spectral coarse space preconditioner.

Abstract

Wave propagation problems are notoriously difficult to solve. Time-harmonic problems are especially challenging in mid and high frequency regimes. The main reason is the oscillatory nature of solutions, meaning that the number of degrees of freedom after discretisation increases drastically with the wave number, giving rise to large complex-valued problems to solve. Additional difficulties occur when the problem is defined in a highly heterogeneous medium, as is often the case in realistic physical applications. For time-discretised problems of Maxwell type, the main challenge remains the significant kernel in curl-conforming spaces, an issue that impacts on the design of robust preconditioners. This has already been addressed theoretically for a homogeneous medium but not yet in the presence of heterogeneities. In this review we provide a big-picture view of the main difficulties encountered when solving wave propagation problems, from the first step of their discretisation through to their parallel solution using two-level methods, by showing their limitations on a few realistic examples. We also propose a new preconditioner inspired by the idea of subspace decomposition, but based on spectral coarse spaces, for curl-conforming discretisations of Maxwell's equations in heterogeneous media.