UNIFORMLY-STABLE FINITE ELEMENT METHODS FOR DARCY-STOKES-BRINKMAN MODELS  

作　　者：Xiaoping Xie Jinchao Xu Guangri Xue 

机构地区：[1]Yangtze Center of Mathematics, and School of Mathematics, Sichuan University, Chengdu 610064, China [2]Center for Computational Mathematics and Applications, Mathematics Department, The Pennsylvania State University, PA 16802, USA

出　　处：《Journal of Computational Mathematics》2008年第3期437-455,共19页计算数学（英文）

基　　金：NSF DMS-0609727;by the Center for Computational Mathematics and Applications of Penn State;Jinchao Xu was also supported in part by NSFC-10501001;Alexander H.Humboldt Foundation.Xiaoping Xie was supported by the National Natural Science Foundation of China (10771150);the National Basic Research Program of China (2005CB321701);the program for New Century Excellent Talents in University (NCET-07-0584)

摘　　要：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 zerothorder 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.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 zerothorder 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.

关 键 词：Darcy-Stokes equation Brinkman Finite element Uniformly stable 

分 类 号：O241.82[理学—计算数学]