Year: 2008
Journal of Computational Mathematics, Vol. 26 (2008), Iss. 3 : pp. 437–455
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 $H(\boldsymbol{Div})$-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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2008-JCM-8635
Journal of Computational Mathematics, Vol. 26 (2008), Iss. 3 : pp. 437–455
Published online: 2008-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 19
Keywords: Darcy-Stokes equation Brinkman Finite element Uniformly stable.