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

Xiaoping Xie, Jinchao Xu, Guangri Xue

Research output: Contribution to journalArticle

76 Citations (Scopus)

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 2008

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Stokes
Stabilization
Finite Element Method
Stokes Equations
Viscosity
Brinkman Equation
Finite element method
Weak Formulation
Divergence-free
Coefficient
Jump
Model
Interface Problems
Darcy's Law
Zeroth
Stokes Problem
Term
Finite Element Approximation
If and only if
Imply

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics
  • Computational Mathematics

Cite this

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Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models. / Xie, Xiaoping; Xu, Jinchao; Xue, Guangri.

In: Journal of Computational Mathematics, Vol. 26, No. 3, 05.2008, p. 437-455.

Research output: Contribution to journalArticle

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