Fully Reliable Localized Error Control in the FEM

If the first task in numerical analysis is the calculation of an approximate solution, the second is to provide a guaranteed error bound and is often of equal importance. The standard approaches in the a posteriori error analysis of finite element methods suppose that the exact solution has a certain regularity or the numerical scheme enjoys some saturation property. For coarse meshes those asymptotic arguments are difficult to recast into rigorous error bounds. The aim of this paper is to provide reliable computable error bounds which are efficient and complete in the sense that constants are estimated as well. The main argument is a localization via a partition of unity which leads to problems on small domains. Two fully reliable estimates are established: The sharper one solves an analytical interface problem with residuals following Babuska and Rheinboldt [SIAM J. Numer. Anal., 15 (1978), pp. 736--754]. The second estimate is a modification of the standard residual-based a posteriori estimate with explicit constants from local analytical eigenvalue problems. For some class of triangulations we show that the efficiency constant is smaller than 2.5. According to our numerical experience, the overestimation of our computable estimates proved to be reasonably small, with an overestimation by a factor between 2.5 and 4 only.