Summary: A POSTERIORI FINITE ELEMENT ERROR ESTIMATION
FOR DIFFUSION PROBLEM
SLIMANE ADJERIDy, BELKACEM BELGUENDOUZz, AND JOSEPH E. FLAHERTYy
Abstract. 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
1 and x
2 in a Carte-
sian x1; x2 frame. Computational results show that the error estimates are accurate, robust, and
e cient 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.
Key words. Finite element methods, a posteriori error estimation, p-re nement, hierarchical
approximations, elliptic and parabolic partial di erential equations.
AMS subject classi cations. 65M60, 65M20, 65M15, 65M50.
1. Introduction. A posteriori error estimates are a standard ingredient of adap-