Superconvergence for rectangular mixed finite elements

SummaryIn this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems. For the rectangular finite elements of Raviart and Thomas [19] and for those of Brezzi et al. [4] we prove that the distance inL2 between the approximate solution and a projection of the exact one is of higher order than the error itself.This result is exploited to obtain superconvergence at Gaussian points and to construct higher order approximations by a local postprocessing.

[1]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[2]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[3]  Ricardo G. Durán.  Error analysis in $L^p \leqslant p \leqslant \infty $ , for mixed finite element methods for linear and quasi-linear elliptic problems , 1988 .

[4]  T. Dupont,et al.  Polynomial approximation of functions in Sobolev spaces , 1980 .

[5]  Ricardo H. Nochetto,et al.  Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations , 1989 .

[6]  D. Arnold,et al.  Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates , 1985 .

[7]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

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

[9]  L. D. Marini,et al.  Two families of mixed finite elements for second order elliptic problems , 1985 .

[10]  Fabio Milner,et al.  Interior and superconvergence estimates for mixed methods for second order elliptic problems , 1985 .

[11]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[12]  Pekka Neittaanmäki,et al.  On superconvergence techniques , 1987 .

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

[14]  M. Zlámal,et al.  Some superconvergence results in the finite element method , 1977 .

[16]  M. Fortin,et al.  E cient rectangular mixed fi-nite elements in two and three space variables , 1987 .

[17]  Michel Fortin,et al.  An analysis of the convergence of mixed finite element methods , 1977 .

[18]  Jr. Jim Douglas.  Improved accuracy through superconvergence in the pressure in the simulation of miscible displacement , 1985 .