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

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.

arXiv comment18 pages, 2 figures, 8 tables

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• A Hybrid High-Order method for the incompressible Navier--Stokes equations based on Temam's device
• A Stable Numerical Algorithm for the Brinkman Equations by Weak Galerkin Finite Element Methods
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Cited by in corpus (5)

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• Pressure-robust staggered DG methods for the Navier-Stokes equations on general meshes
• A pressure robust staggered discontinuous Galerkin method for the Stokes equations
• A uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem
• Locking free staggered DG method for the Biot system of poroelasticity on general polygonal meshes