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Volume 26, Issue 3
Uniformly Stable Finite Element Methods for Darcy-Stokes-Brinkman Models

Xiaoping Xie, Jinchao Xu & Guangri Xue

J. Comp. Math., 26 (2008), pp. 437-455.

Published online: 2008-06

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  • 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.

  • AMS Subject Headings

65N12, 65N15, 65N22, 65N30.

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COPYRIGHT: © Global Science Press

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@Article{JCM-26-437, author = {Xiaoping Xie, Jinchao Xu and Guangri Xue}, title = {Uniformly Stable Finite Element Methods for Darcy-Stokes-Brinkman Models}, journal = {Journal of Computational Mathematics}, year = {2008}, volume = {26}, number = {3}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8635.html} }
TY - JOUR T1 - Uniformly Stable Finite Element Methods for Darcy-Stokes-Brinkman Models AU - Xiaoping Xie, Jinchao Xu & Guangri Xue JO - Journal of Computational Mathematics VL - 3 SP - 437 EP - 455 PY - 2008 DA - 2008/06 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8635.html KW - Darcy-Stokes equation, Brinkman, Finite element, Uniformly stable. AB -

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.

Xiaoping Xie, Jinchao Xu and Guangri Xue. (2008). Uniformly Stable Finite Element Methods for Darcy-Stokes-Brinkman Models. Journal of Computational Mathematics. 26 (3). 437-455. doi:
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