A strongly mass conservative method for the coupled Brinkman-Darcy flow and transport

TL;DR

This paper introduces a novel, strongly mass conservative, and stabilizer-free numerical scheme for coupled Brinkman-Darcy flow and transport, ensuring robustness, accuracy, and natural interface condition incorporation without additional variables.

Contribution

The paper presents a new strongly mass conservative scheme for coupled flow and transport that is stabilizer-free, pressure-robust, and naturally incorporates interface conditions.

Findings

Scheme is robust for various viscosities

Velocity error is pressure-independent

Numerical experiments confirm theoretical results

Abstract

In this paper, a strongly mass conservative and stabilizer free scheme is designed and analyzed for the coupled Brinkman-Darcy flow and transport. The flow equations are discretized by using a strongly mass conservative scheme in mixed formulation with a suitable incorporation of the interface conditions. In particular, the interface conditions can be incorporated into the discrete formulation naturally without introducing additional variables. Moreover, the proposed scheme behaves uniformly robust for various values of viscosity. A novel upwinding staggered DG scheme in mixed form is exploited to solve the transport equation, where the boundary correction terms are added to improve the stability. A rigorous convergence analysis is carried out for the approximation of the flow equations. The velocity error is shown to be independent of the pressure and thus confirms the pressure-robustness. Stability and a priori error estimates are also obtained for the approximation of the transport equation; moreover, we are able to achieve a sharp stability and convergence error estimates thanks to the strong mass conservation preserved by our scheme. In particular, the stability estimate depends only on the true velocity on the inflow boundary rather than on the approximated velocity. Several numerical experiments are presented to verify the theoretical findings and demonstrate the performances of the method.