 
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 bip 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 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 quadrilateralelement meshes, problems with singularities, and nonlinear problems.
Key words. Finite element methods, a posteriori error estimation, pre 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
