Hybridizable discontinuous Galerkin methods for the coupled Stokes--Biot problem

TL;DR

This paper introduces a hybridizable discontinuous Galerkin method for the coupled Stokes--Biot problem that ensures local conservation, is locking-free, and achieves optimal convergence rates.

Contribution

The paper develops a novel HDG finite element method for the Stokes--Biot problem with $H( ext{div})$-conforming velocities and displacements, ensuring strong conservation and locking-free performance.

Findings

Method achieves optimal convergence rates in $L^2$-norm.

Discretization is locking-free in the incompressible limit.

Numerical examples confirm theoretical error estimates.

Abstract

We present and analyze a hybridizable discontinuous Galerkin (HDG) finite element method for the coupled Stokes--Biot problem. Of particular interest is that the discrete velocities and displacement are $H(\text{div})$-conforming and satisfy the compressibility equations pointwise on the elements. Furthermore, in the incompressible limit, the discretization is strongly conservative. We prove well-posedness of the discretization and, after combining the HDG method with backward Euler time stepping, present a priori error estimates that demonstrate that the method is free of volumetric locking. Numerical examples further demonstrate optimal rates of convergence in the $L^2$-norm for all unknowns and that the discretization is locking-free.