A unified approach to a posteriori error estimation based on duality error majorants

In this paper, we present a unified approach to computing sharp error estimates for approximate solutions of elliptic type boundary value problems in variational form. This approach is based on the duality theory of the calculus of variations which analyses the original (primal) variational problem together with another (dual) one. In previous papers this theory was used to obtain a new tool of a posteriori error estimation – duality error majorant (DEM). The latter provides an upper bound for the energy norm of the error for any approximation that belongs to the set of admissible functions of the considered variational problem. For this reason, DEM can be easily used to estimate the accuracy of various post-processed finite element approximations as well as to those computed by boundary element or by finite difference methods. The objective of this paper is to present practically convenient forms of DEM and to discuss computational aspects of this error estimation strategy. The performance of the proposed error estimation method is demonstrated through several examples, where duality majorants are computed and compared with the results obtained by other methods.

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