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 disconti...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.展开更多
In this paper,we apply streamline-diffusion and Galerkin-least-squares fi-nite element methods for 2D steady-state two-phase model in the cathode of polymer electrolyte fuel cell(PEFC)that contains a gas channel and a...In this paper,we apply streamline-diffusion and Galerkin-least-squares fi-nite element methods for 2D steady-state two-phase model in the cathode of polymer electrolyte fuel cell(PEFC)that contains a gas channel and a gas diffusion layer(GDL).This two-phase PEFC model is typically modeled by a modified Navier-Stokes equation for the mass and momentum,with Darcy’s drag as an additional source term in momentum for flows through GDL,and a discontinuous and degenerate convectiondiffusion equation for water concentration.Based on the mixed finite element method for the modified Navier-Stokes equation and standard finite element method for water equation,we design streamline-diffusion and Galerkin-least-squares to overcome the dominant convection arising from the gas channel.Meanwhile,we employ Kirchhoff transformation to deal with the discontinuous and degenerate diffusivity in water concentration.Numerical experiments demonstrate that our finite element methods,together with these numerical techniques,are able to get accurate physical solutions with fast convergence.展开更多
基金NSF DMS-0609727by the Center for Computational Mathematics and Applications of Penn State+3 种基金Jinchao Xu was also supported in part by NSFC-10501001Alexander 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)
文摘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.
基金This work was supported in part by NSF DMS-0609727,the Center for Computa-tional Mathematics and Applications of Penn State University.J.Xu was also supported in part by NSFC-10501001 and Alexander H.Humboldt Foundation.
文摘In this paper,we apply streamline-diffusion and Galerkin-least-squares fi-nite element methods for 2D steady-state two-phase model in the cathode of polymer electrolyte fuel cell(PEFC)that contains a gas channel and a gas diffusion layer(GDL).This two-phase PEFC model is typically modeled by a modified Navier-Stokes equation for the mass and momentum,with Darcy’s drag as an additional source term in momentum for flows through GDL,and a discontinuous and degenerate convectiondiffusion equation for water concentration.Based on the mixed finite element method for the modified Navier-Stokes equation and standard finite element method for water equation,we design streamline-diffusion and Galerkin-least-squares to overcome the dominant convection arising from the gas channel.Meanwhile,we employ Kirchhoff transformation to deal with the discontinuous and degenerate diffusivity in water concentration.Numerical experiments demonstrate that our finite element methods,together with these numerical techniques,are able to get accurate physical solutions with fast convergence.
