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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 = {}, 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 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 & Guangri Xue. (1970). Uniformly Stable Finite Element Methods for Darcy-Stokes-Brinkman Models. Journal of Computational Mathematics. 26 (3). 437-455. doi:
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