Influence of the SIPG penalisation on the numerical properties of linear systems for elastic wave propagation

TL;DR

This paper investigates how the SIPG penalisation affects the numerical properties like condition number and eigenvalue distribution of linear systems derived from space-time discretisations in elastic wave propagation, aiming to improve solver robustness.

Contribution

It provides a numerical study of the influence of SIPG penalisation on the properties of linear systems for elastic wave problems, highlighting dependencies on penalisation and time interval length.

Findings

Condition number varies with penalisation parameters.

Eigenvalue distribution depends on discretisation choices.

Penalisation impacts the efficiency of iterative solvers.

Abstract

Interior penalty discontinuous Galerkin discretisations (IPDG) and especially the symmetric variant (SIPG) for time-domain wave propagation problems are broadly accepted and widely used due to their advantageous properties. Linear systems with block structure arise by applying space-time discretisations and reducing the global system to time-slab problems. The design of efficient and robust iterative solvers for linear systems from interior penalty discretisations for hyperbolic wave equations is still a challenging task and relies on understanding the properties of the systems. In this work the numerical properties such as the condition number and the distribution of eigenvalues of different representations of the linear systems coming from space-time discretisations for elastic wave propagation are numerically studied. These properties for interior penalty discretisations depend on the penalisation and on the time interval length.