Extrapolation of the Nédélec element for the Maxwell equations by the mixed finite element method

In this paper, we use the integral-identity argument to obtain asymptotic error expansions for the mixed finite element approximation of the Maxwell equations on a rectangular mesh. The extrapolation method is applied to improve the accuracy of the approximation via an interpolation postprocessing technique. With the extrapolation, the approximation accuracy can be improved from O(h) to O(h4) in the L2-norm. Illustrative numerical results are given to demonstrate the higher order accuracy of the extrapolation method.

[1]  Peter Monk,et al.  Analysis of a finite element method for Maxwell's equations , 1992 .

[2]  Peter Monk,et al.  Superconvergence of finite element approximations to Maxwell's equations , 1994 .

[3]  Qun Lin,et al.  Global superconvergence for Maxwell's equations , 2000, Math. Comput..

[4]  Jan Brandts,et al.  Superconvergence of mixed finite element semi-discretizations of two time-dependent problems , 1999 .

[5]  Aihui Zhou,et al.  Global superconvergence approximations of the mixed finite element method for the Stokes problem and the linear elasticity equation , 1996 .

[6]  Shuhua Zhang,et al.  Asymptotic Expansions and Richardson Extrapolation of Approximate Solutions for Second Order Elliptic Problems on Rectangular Domains by Mixed Finite Element Methods , 2006, SIAM J. Numer. Anal..

[7]  Jia-fu Lin,et al.  Extrapolation of the Hood–Taylor elements for the Stokes problem , 2005, Adv. Comput. Math..

[8]  Jinchao Xu,et al.  Asymptotically Exact A Posteriori Error Estimators, Part I: Grids with Superconvergence , 2003, SIAM J. Numer. Anal..

[9]  Qunlin,et al.  GLOBAL SUPERCONVERGENCE OF THE MIXED FINITEELEMENT METHODS FOR 2-D MAXWELL EQUATIONS , 2003 .

[10]  Jinchao Xu,et al.  Asymptotically Exact A Posteriori Error Estimators, Part II: General Unstructured Grids , 2003, SIAM J. Numer. Anal..

[11]  R. S. Falk,et al.  Error estimates for mixed methods , 1980 .

[12]  Peter Monk,et al.  A mixed method for approximating Maxwell's equations , 1991 .

[13]  Leszek Demkowicz,et al.  Finite Element Methods for Maxwell Equations , 2007 .

[14]  Peter Monk.  A Comparison of Three Mixed Methods for the Time-Dependent Maxwell's Equations , 1992, SIAM J. Sci. Comput..

[15]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[16]  Qun Lin,et al.  EXTRAPOLATION AND DEFECT CORRECTION FOR DIFFUSION EQUATIONS WITH BOUNDARY INTEGRAL CONDITIONS , 1997 .

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

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

[19]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[20]  Jun Ping Wang.  Superconvergence and extrapolation for mixed finite element methods on rectangular domains , 1991 .