Local a posteriori estimates for pointwise gradient errors in finite element methods for elliptic problems, Preprint, Fakultät für Mathematik und Physik, Universität Freiburg

Abstract. We prove local a posteriori error estimates for pointwise gradi-ent errors in finite element methods for a second-order linear elliptic model problem. First we split the local gradient error into a computable local resid-ual term and a weaker global norm of the finite element error (the “pollution term”). Using a mesh-dependent weight, the residual term is bounded in a sharply localized fashion. In specific situations the pollution term may also be bounded by computable residual estimators. On nonconvex polygonal and polyhedral domains in two and three space dimensions, we may choose estima-tors for the pollution term which do not employ specific knowledge of corner singularities and which are valid on domains with cracks. The finite element mesh is only required to be simplicial and shape-regular, so that highly graded and unstructured meshes are allowed