Energy-corrected finite element methods for corner singularities

It is well known that the regularity of solutions o f elliptic partial differential equations on domains with re-entrant corners is limited by the maximal interior angle. This results in reduced convergence rates for finite element approximations on families of quasi-uniform meshes. Following an idea of Zenger and Gietl, we show that it is possible to regain the full order of convergence by a local modification of the bilinear form in a vicinity of the singularity and thus to overcome the pollution effect. A complete convergence analysis in weighted Sobolev spaces is presented, and we also show that the stress intensity factors can be computed with optimal accuracy. The theoretical results are illustrated by numerical tests that demonstrate second order convergence of linear finite elements on families of quasi-uniform meshes.