A New Time Discontinuous Expanded Mixed Element Method for Convection-dominated Diffusion Equation

In this paper, a new time discontinuous expanded mixed finite element method is proposed and analyzed for two-order convection-dominated diffusion problem. The proofs of the stability of the proposed scheme and the uniqueness of the discrete solution are given. Moreover, the error estimates of the scalar unknown, its gradient and its flux in the L∞(J , L(Ω)-norm are obtained. Keywords—Convection-dominated diffusion equation; Expanded mixed method; Time discontinuous scheme; Stability; Error estimates.

[1]  Fabio Milner,et al.  Mixed finite element methods for quasilinear second-order elliptic problems , 1985 .

[2]  Carol S. Woodward,et al.  Analysis of Expanded Mixed Finite Element Methods for a Nonlinear Parabolic Equation Modeling Flow into Variably Saturated Porous Media , 2000, SIAM J. Numer. Anal..

[3]  Hongxing Rui,et al.  An expanded mixed covolume method for elliptic problems , 2005 .

[4]  Yunqing Huang,et al.  A two‐grid method for expanded mixed finite‐element solution of semilinear reaction–diffusion equations , 2003 .

[5]  Zhangxin Chen,et al.  Expanded mixed finite element methods for linear second-order elliptic problems, I , 1998 .

[6]  Li Hong.  A New Mixed Finite Element Method for Pseudo-Hyperbolic Equation , 2010 .

[7]  Chen Huan-zhen.  Mixed finite element method for the convection-dominated diffusion problems with small parameterε , 1998 .

[8]  A. Ware.  A spectral Lagrange-Galerkin method for convection-dominated diffusion problems , 1994 .

[9]  Hong Wang,et al.  An optimal‐order error estimate on an H1‐Galerkin mixed method for a nonlinear parabolic equation in porous medium flow , 2010 .

[10]  Gunar Matthies,et al.  A streamline-diffusion method for nonconforming finite element approximations applied to convection-diffusion problems , 1998 .

[11]  Chen,et al.  AN EXPANDED CHARACTERISTIC-MIXED FINITE ELEMENT METHOD FOR A CONVECTION-DOMINATED TRANSPORT PROBLEM , 2005 .

[12]  Todd Arbogast,et al.  Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Finite Differences , 1994 .

[13]  Zhangxin Chen.  Analysis of expanded mixed methods for fourth‐order elliptic problems , 1997 .

[14]  Yang Liu,et al.  Mixed time discontinuous space-time finite element method for convection diffusion equations , 2008 .

[15]  A. Peirce.  Computer Methods in Applied Mechanics and Engineering , 2010 .

[16]  Dongho Kim,et al.  A posteriori error estimator for expanded mixed hybrid methods , 2007 .

[17]  Haitao Che,et al.  An Optimal Error Estimates of H1-Galerkin Expanded Mixed Finite Element Methods for Nonlinear Viscoelasticity-Type Equation , 2011 .

[18]  Ziwen Jiang,et al.  Expanded mixed finite element methods for the problem of purely longitudinal motion of a homogeneous bar , 2011, J. Comput. Appl. Math..

[19]  Wei Liu,et al.  A two-grid method with expanded mixed element for nonlinear reaction-diffusion equations , 2011 .

[20]  Yirang Yuan,et al.  The expanded upwind-mixed multi-step method for the miscible displacement problem in three dimensions , 2008, Appl. Math. Comput..

[21]  T. F. Russell,et al.  NUMERICAL METHODS FOR CONVECTION-DOMINATED DIFFUSION PROBLEMS BASED ON COMBINING THE METHOD OF CHARACTERISTICS WITH FINITE ELEMENT OR FINITE DIFFERENCE PROCEDURES* , 1982 .