A tutorial in elementary finite element error analysis: A systematic presentation of a priori and a posteriori error estimates Original Research Article

This paper provides a tutorial on the derivation of a priori and explicit residual-based a posteriori error estimates for Galerkin finite element discretizations of general linear elliptic operators. For simplicity, the presentation is in one dimension, although extensions to multidimensions should be straightforward. The structures of explicit residual-based a posteriori error estimators and a priori error estimators 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 discuss a priori and a posteriori error estimators in parallel. In this paper error bounds are obtained for the error measured in the energy norm and the L2 norm. In addition, symmetric, nonsymmetric, positive definite and indefinite operators are discussed. A priori estimates for nonsymmetric operators are treated with a novel approach involving the introduction of the skew norm. Special attention is given to the advection-diffusion equation (a nonsymmetric operator) and the Helmholtz equation (an indefinite operator). We systematically describe the necessary steps in deriving a priori and a posteriori error estimates, and provide a simplified understanding of their differences as well as their similarities.