Low Order Nonconforming Expanded Characteristic- Mixed Finite Element Method for the Convection- Diffusion Problem

A low order nonconforming finite element method is proposed for the convection-diffusion equations with the expanded characteristic-mixed finite element scheme. The method is a combination of characteristic approximation to handle the convection part in time and a expanded nonconforming mixed finite element spatial approximation to deal with the diffusion part. In the process, the interpolation operator is employed instead of the so-called elliptic projection which is an indispensable tool used for the convergence analysis in the previous literature. When the exact solutions belong to H 2 (Ω) instead of H 3 (Ω), the corresponding optimal order error estimates in L 2 -norm are obtained by use of some distinct properties of the nonconforming finite elements.

[1]  Dongyang Shi,et al.  A new low-order non-conforming mixed finite-element scheme for second-order elliptic problems , 2011, Int. J. Comput. Math..

[2]  Shi Dong-yang,et al.  A new nonconforming mixed finite element scheme for the stationary Navier-Stokes equations , 2011 .

[3]  Shao-chunChen,et al.  AN ANISOTROPIC NONCONFORMING FINITE ELEMENT WITH SOME SUPERCONVERGENCE RESULTS , 2005 .

[4]  Lutz Tobiska,et al.  The streamline–diffusion method for nonconforming Qrot1 elements on rectangular tensor–product meshes , 2001 .

[5]  Jean E. Roberts,et al.  Global estimates for mixed methods for second order elliptic equations , 1985 .

[6]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[7]  T. F. Russell,et al.  Time Stepping Along Characteristics with Incomplete Iteration for a Galerkin Approximation of Miscible Displacement in Porous Media , 1985 .

[8]  Yirang Yuan,et al.  The characteristic finite volume element method for the nonlinear convection-dominated diffusion problem , 2008, Comput. Math. Appl..

[9]  Dongwoo Sheen,et al.  P1-Nonconforming Quadrilateral Finite Element Methods for Second-Order Elliptic Problems , 2003, SIAM J. Numer. Anal..

[10]  Shi,et al.  CONSTRAINED QUADRILATERAL NONCONFORMING ROTATED Q1 ELEMENT , 2005 .

[11]  Mary F. Wheeler,et al.  Some improved error estimates for the modified method of characteristics , 1989 .

[12]  Dongyang Shi,et al.  LOW ORDER CROUZEIX-RAVIART TYPE NONCONFORMING FINITE ELEMENT METHODS FOR APPROXIMATING MAXWELL'S EQUATIONS , 2008 .

[13]  Ningning Yan,et al.  Superconvergence analysis of the nonconforming quadrilateral linear-constant scheme for Stokes equations , 2008, Adv. Comput. Math..

[14]  Dan-Ping Yang,et al.  Analysis of least-squares mixed finite element methods for nonlinear nonstationary convection-diffusion problems , 2000, Math. Comput..

[15]  T. F. Russell,et al.  Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics , 1984 .

[16]  Dong-Yang Shi,et al.  Two low order characteristic finite element methods for a convection-dominated transport problem , 2010, Comput. Math. Appl..

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

[18]  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 .