Fully reliable localized error control in the FEM

If the first task in numerical analysis is the calculation of an approximat e solution, the second is to provide a guaranteed error bound and is often of equal importance. The standard approaches in the a posteriori error anal ysis of finite element methods suppose that the exact solution has a certai n regularity or the numerical scheme enjoys some saturation property. For c oarse meshes those asymptotic arguments are difficult to recast into rigoro us error bounds. The aim of this paper is to provide reliable computable er ror 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 estimate s are established. The sharper one solves an analytical interface problem w ith residuals following Babuska and Rheinboldt [SIAM J. Numer. Anal., 15 (1 978), pp. 736-754]. The second estimate is a modification of the standard r esidual-based a posteriori estimate with explicit constants from local anal ytical eigenvalue problems. For some class of triangulations we show that t he efficiency constant is smaller than 2.5. According to our numerical expe rience, the overestimation of our computable estimates proved to be reasona bly small, with an overestimation by a factor between 2.5 and 4 only.