Higher-order gradient post-processings for second-order elliptic problems

Abstract Global, element-by-element and macroelement post-processing recovery techniques based on least-square residuals of equilibrium equation and irrotationality condition are proposed for second-order elliptic problems. Improved accuracy for the flux finite element approximations is obtained with low computational cost and easy implementation. Error estimates are derived and numerical experiments are reported confirming the higher-order rates of convergence predicted in the analysis.

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

[2]  Abimael F. D. Loula,et al.  Improved accuracy in finite element analysis of Biot's consolidation problem , 1992 .

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

[4]  R. Stenberg.  Analysis of mixed finite elements methods for the Stokes problem: a unified approach , 1984 .

[5]  J. Barlow,et al.  Optimal stress locations in finite element models , 1976 .

[6]  M. Zlámal,et al.  Superconvergence of the gradient of finite element solutions , 1979 .

[7]  Miloš Zlámal,et al.  Superconvergence and reduced integration in the finite element method , 1978 .

[8]  J. N. Reddy,et al.  Note on an approximate method for computing consistent conjugate stresses in elastic finite elements , 1973 .

[9]  Long‐time behavior of some Galerkin and Petrov–Galerkin methods for thermoelastic consolidation , 1995 .

[10]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[11]  Nils-Erik Wiberg,et al.  Patch recovery based on superconvergent derivatives and equilibrium , 1993 .

[12]  J. Douglas,et al.  PEERS: A new mixed finite element for plane elasticity , 1984 .

[13]  Richard E. Ewing,et al.  The approximation of the pressure by a mixed method in the simulation of miscible displacement , 1983 .

[14]  J. Bramble,et al.  Higher order local accuracy by averaging in the finite element method , 1977 .

[15]  Ted Belytschko,et al.  Superconvergent patch recovery with equilibrium and conjoint interpolant enhancements , 1994 .

[16]  Nils-Erik Wiberg,et al.  Superconvergent patch recovery of finite‐element solution and a posteriori L2 norm error estimate , 1994 .

[17]  R. Stenberg.  A family of mixed finite elements for the elasticity problem , 1988 .

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

[19]  R. Ewing.  Simulation of Multiphase Flows in Porous Media , 1991 .

[20]  D. Arnold,et al.  A new mixed formulation for elasticity , 1988 .

[21]  T. Belytschko,et al.  Enhanced derivative recovery through least square residual penalty , 1994 .

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

[23]  John S. Campbell,et al.  Local and global smoothing of discontinuous finite element functions using a least squares method , 1974 .