A Posteriori Finite Element Error Estimation for Diffusion Problems

Adjerid et al. 2] and Yu 19, 20] show that a posteriori estimates of spatial discretiza-tion errors of piecewise bi-p polynomial nite element solutions of elliptic and parabolic problems on meshes of square elements may be obtained from jumps in solution gradients at element vertices when p is odd and from local elliptic or parabolic problems when p is even. We show that these simple error estimates are asymptotically correct for other nite element spaces. The key requirement is that the trial space contain all monomial terms of degree p + 1 except for x p+1 1 and x p+1 2 in a Carte-sian (x 1 ; x 2) frame. Computational results show that the error estimates are accurate, robust, and eecient for a wide range of problems, including some that are not supported by the present theory. These involve quadrilateral-element meshes, problems with singularities, and nonlinear problems.