A hybridizable discontinuous Galerkin method for the fully coupled time-dependent Stokes/Darcy-transport problem

TL;DR

This paper introduces a high-order hybridized discontinuous Galerkin method for solving the fully coupled time-dependent Stokes-Darcy-transport problem, ensuring mass conservation, interface compatibility, and robustness through a decoupling strategy.

Contribution

The paper develops a novel HDG method that guarantees strong mass conservation, handles interface conditions naturally, and provides optimal error estimates for coupled flow and transport problems.

Findings

Method achieves optimal error estimates.

Numerical examples confirm robustness and compatibility.

Discretization respects interface conditions.

Abstract

We present a high-order hybridized discontinuous Galerkin (HDG) method for the fully coupled time-dependent Stokes-Darcy-transport problem where the fluid viscosity and source/sink terms depend on the concentration and the dispersion/diffusion tensor depends on the fluid velocity. This HDG method is such that the discrete flow equations are compatible with the discrete transport equation. Furthermore, the HDG method guarantees strong mass conservation in the $H^{\rm div}$ sense and naturally treats the interface conditions between the Stokes and Darcy regions via facet variables. We employ a linearizing decoupling strategy where the Stokes/Darcy and the transport equations are solved sequentially by time-lagging the concentration. We prove well-posedness and optimal a priori error estimates for the velocity and the concentration in the energy norm. We present numerical examples that respect compatibility of the flow and transport discretizations and demonstrate that the discrete solution is robust with respect to the problem parameters.