Locally Efficient and Reliable a Posteriori Error Estimators for Dirichlet Problems

Considering the Dirichlet problem for Poisson's equation in two and three dimensions, we derive a posteriori error estimators for finite element solutions with interpolated boundary values. The estimators are reliable and (locally) efficient with respect to the energy norm error, also in the case of discontinuous boundary values and load terms that are not square-integrable due to singularities at the boundary of the underlying domain. Moreover, we propose an adaptive algorithm based upon these estimators and test it also in nonsmooth cases of the aforementioned type: its convergence rate is optimal.