A new staggered DG method for the Brinkman problem robust in the Darcy and Stokes limits

TL;DR

This paper introduces a novel staggered discontinuous Galerkin method for the Brinkman problem that is robust across Darcy and Stokes flow regimes, allowing for automatic handling of hanging nodes and achieving optimal error estimates.

Contribution

The paper develops a new staggered DG method for the Brinkman problem with relaxed velocity continuity and proven superconvergence, improving robustness and accuracy over existing methods.

Findings

Method is robust in Stokes and Darcy limits.

Achieves optimal $L^2$ error estimates independent of viscosity.

Numerical results confirm theoretical accuracy and performance.

Abstract

In this paper we propose a novel staggered discontinuous Galerkin method for the Brinkman problem on general quadrilateral and polygonal meshes. The proposed method is robust in the Stokes and Darcy limits, in addition, hanging nodes can be automatically incorporated in the construction of the method, which are desirable features in practical applications. There are three unknowns involved in our formulation, namely velocity gradient, velocity and pressure. Unlike the original staggered DG formulation proposed for the Stokes equations in \cite{KimChung13}, we relax the tangential continuity of velocity and enforce different staggered continuity properties for the three unknowns, which is tailored to yield an optimal $L^2$ error estimates for velocity gradient, velocity and pressure independent of the viscosity coefficient. Moreover, by choosing suitable projection, superconvergence can be proved for $L^2$ error of velocity. Finally, several numerical results illustrating the good performances of the proposed method and confirming the theoretical findings are presented.