On mathematical and numerical modelling of multiphysics wave propagation with polytopal Discontinuous Galerkin methods

TL;DR

This paper reviews and analyzes polytopal Discontinuous Galerkin methods for simulating complex multiphysics wave phenomena in heterogeneous media, including elastic, poro-elastic, and acoustic materials, with stability and error estimates.

Contribution

It introduces and analyzes a semi-discrete PolydG formulation for multiphysics wave models, including stability and error analysis, and validates the methods with numerical tests.

Findings

Stable semi-discrete formulation for multiphysics wave models

Error estimates in energy norm for the proposed methods

Validation through numerical tests with manufactured solutions

Abstract

In this work we review discontinuous Galerkin finite element methods on polytopal grids (PolydG) for the numerical simulation of multiphysics wave propagation phenomena in heterogeneous media. In particular, we address wave phenomena in elastic, poro-elastic, and poro-elasto-acoustic materials. Wave propagation is modeled by using either the elastodynamics equation in the elastic domain, the acoustics equations in the acoustic domain and the low-frequency Biot's equations in the poro-elastic one. The coupling between different models is realized by means of (physically consistent) transmission conditions, weakly imposed at the interface between the subdomains. For all models configuration, we introduce and analyse the PolydG semi-discrete formulation, which is then coupled with suitable time marching schemes. For the semi-discrete problem, we present the stability analysis and derive a-priori error estimates in a suitable energy norm. A wide set of verification tests with manufactured solutions are presented in order to validate the error analysis. Examples of physical interest are also shown to demonstrate the capability of the proposed methods.