Aspects of Guaranteed Error Control in CPDEs

Abstract

Whenever numerical algorithms are employed for a reliable computational forecast, they need to allow for an error control in the final quantity of interest. The discretization error control is of some particular importance in computational PDEs (CPDEs) where guaranteed upper error bound (GUB) are of vital relevance. After a quick overview over energy norm error control in second-order elliptic PDEs, this paper focuses on three particular aspects: first, the variational crimes from a nonconforming finite element discretization and guaranteed error bounds in the discrete norm with improved postprocessing of the GUB; second, the reliable approximation of the discretization error on curved boundaries; and finally, the reliable bounds of the error with respect to some goal functional, namely, the error in the approximation of the directional derivative at a given point.