Derivation of fully computable error bounds from a posteriori error identities

S. Repin

Zapiski Nauchnykh Seminarov POMI, Investigations on applied mathematics and informatics. Part III, Tome 539 (2024), pp. 120-156

Résumé

A posteriori error identities are functional relations that control distances between the exact solution of a boundary value problem and any function from the respective energy space. They have been derived for many boundary value problems associated with partial differential equations of elliptic and parabolic types. A posteriori identities have a common structure: their left hand sides form certain error measures and the right hand ones consist of directly computable terms and a linear functional, which contains unknown error function. Fully computable estimates follow from such an identity provided that this functional is efficiently estimated. The difficulty that arises is due to the fact that computational simplicity and efficiency of such an estimate are contradictory requirements. A method suggested in the paper, largely overcomes this difficulty. It uses an auxiliary finite dimensional problem to estimate the linear functional containing unknown error function. The resulting estimates minimise possible overestimation of this term and imply sharp and fully computable majorants and minorants of errors.