Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models

Xiaoping Xie, Jinchao Xu, Guangri Xue

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89 Scopus citations

Abstract

In this paper, we consider 2D and 3D Darcy-Stokes interface problems. These equations are related to Brinkman model that treats both Darcy's law and Stokes equations in a single form of PDE but with strongly discontinuous viscosity coefficient and zeroth-order term coefficient. We present three different methods to construct uniformly stable finite element approximations. The first two methods are based on the original weak formulations of Darcy-Stokes-Brinkman equations. In the first method we consider the existing Stokes elements. We show that a stable Stokes element is also uniformly stable with respect to the coefficients and the jumps of Darcy-Stokes-Brinkman equations if and only if the discretely divergence-free velocity implies almost everywhere divergence-free one. In the second method we construct uniformly stable elements by modifying some well-known H (div)-conforming elements. We give some new 2D and 3D elements in a unified way. In the last method we modify the original weak formulation of Darcy-Stokes-Brinkman equations with a stabilization term. We show that all traditional stable Stokes elements are uniformly stable with respect to the coefficients and their jumps under this new formulation.

Original languageEnglish (US)
Pages (from-to)437-455
Number of pages19
JournalJournal of Computational Mathematics
Volume26
Issue number3
StatePublished - May 1 2008

All Science Journal Classification (ASJC) codes

  • Computational Mathematics

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