A compatible embedded-hybridized discontinuous Galerkin method for the Stokes--Darcy-transport problem

TL;DR

This paper introduces a stable and compatible embedded-hybridized discontinuous Galerkin method for coupled Stokes--Darcy flow and transport, providing theoretical analysis and numerical validation of its effectiveness.

Contribution

The paper develops a new compatible EDG-HDG finite element method for coupled flow and transport problems, with stability, error estimates, and comparison to incompatible discretizations.

Findings

The compatible EDG-HDG method is stable and free of spurious oscillations.

Optimal a priori error estimates are established.

Numerical examples confirm theoretical results and demonstrate improved solution quality.

Abstract

We present a stability and error analysis of an embedded-hybridized discontinuous Galerkin (EDG-HDG) finite element method for coupled Stokes--Darcy flow and transport. The flow problem, governed by the Stokes--Darcy equations, is discretized by a recently introduced exactly mass conserving EDG-HDG method while an embedded discontinuous Galerkin (EDG) method is used to discretize the transport equation. We show that the coupled flow and transport discretization is compatible and stable. Furthermore, we show existence and uniqueness of the semi-discrete transport problem and develop optimal a priori error estimates. We provide numerical examples illustrating the theoretical results. In particular, we compare the compatible EDG-HDG discretization to a discretization of the coupled Stokes--Darcy and transport problem that is not compatible. We demonstrate that where the incompatible discretization may result in spurious oscillations in the solution to the transport problem, the compatible discretization is free of oscillations. An additional numerical example with realistic parameters is also presented.