Hybridizable discontinuous Galerkin methods for poroelastic wave propagation with symmetric stress approximation

TL;DR

This paper introduces hybridized discontinuous Galerkin methods for poroelastic wave equations, emphasizing symmetric stress approximation, error robustness, and efficient system condensation, supported by theoretical analysis and numerical validation.

Contribution

The paper develops a novel HDG framework combining two approaches for poroelastic waves, achieving symmetric stress approximation and robust error convergence.

Findings

Achieves optimal convergence in numerical examples.

Maintains symmetric stress approximation in discretization.

System condensation simplifies computational complexity.

Abstract

In this paper, we develop hybridized discontinuous Galerkin (HDG) methods for poroelastic wave equations. We first rewrite the governing equations to a first-order symmetric hyperbolic system in order to use dual mixed formulations for discretization. Subsequently, we combine two HDG approaches in the discretization of the system, the $\text{HDG}+$ method for the linear elasticity equations and the $\text{LDG-H}$ method for the diffusion equations, with adjustments for the poroelastic wave equations. In our proposed HDG methods, the numerical approximation of the stress tensor is strongly symmetric and the convergence of the errors are robust for nearly incompressible materials. Upon performing static condensation, the system retains numerical trace variables solely for the solid displacement and the fluid pressure. We provide comprehensive error analyses for both the semidiscrete formulation and the Crank--Nicolson time-stepping scheme. Finally, extensive numerical examples illustrate optimal convergence results and simulate different poroelastic wave propagation scenarios relevant in the literature.