A TUTORIAL IN ELEMENTARY FINITE-ELEMENT ERROR ANALYSIS - A SYSTEMATICPRESENTATION OF A-PRIORI AND A-POSTERIORI ERROR-ESTIMATES

This paper provides a tutorial on the derivation of a priori and expli cit residual-based a posteriori error estimates for Galerkin finite el ement discretizations of general linear elliptic operators. For simpli city, the presentation is in one dimension, although extensions to mul tidimensions should be straightforward. The structures of explicit res idual-based a posteriori error estimators and a priori error estimator s are very similar-they are each in the form of an upper bound on the error measured in some norm. The structures of their derivations also follow similar lines. It is therefore appropriate to present and discu ss a priori and a posteriori error estimators in parallel. In this pap er error bounds are obtained for the error measured in the energy norm and the L-2 norm. In addition, symmetric, nonsymmetric, positive defi nite and indefinite operators are discussed. A priori estimates for no nsymmetric operators are treated with a novel approach involving the i ntroduction of the skew norm. Special attention is given to the advect ion-diffusion equation (a nonsymmetric operator) and the Helmholtz equ ation (an indefinite operator). We systematically describe the necessa ry steps in deriving a priori and a posteriori error estimates, and pr ovide a simplified understanding of their differences as well as their similarities. (C) 1998 Elsevier Science S.A.