Fully reliable error control in the h-p-version of FEM

Some approaches in the a posteriori error analysis of finite element method s (FEM) are based on the regularity of the exact solution or on a saturatio n property of the numerical scheme. For coarse meshes those asymptotic argu ments are difficult to recast into rigorous error bounds. Here, we will pro vide reliable computable error bounds which are efficient and complete in t he sense that constants are estimated as well. A localisation via a partiti on of unity yields problems an small domains. Two fully reliable estimates are established, the sharper one solves an analytical interface problem wit h residuals following Babuska and Rheinboldt. The second estimate yields a modification of the standard residual-based a posteriori estimate with expl icit constants computed from local analytical eigenvalue problems. Emphasis is on the efficiency of the computed error bound, which can be monitored. For some class of triangulations and the h-version we show that the efficie ncy constant is smaller than 2.5 and grows only weakly for the h-p-version. Numerical experiments support and illustrate the theoretical results.