A priori mesh grading for an elliptic problem with dirac right-hand side

The Green function of the Poisson equation in two dimensions is not contained in the Sobolev space H¹(Ω) such that finite element error estimates for the discretization of a problem with the Dirac measure on the right hand-side are nonstandard and quasi-uniform meshes are inappropriate. By using graded meshes L²-error estimates of almost optimal order are shown. As a byproduct, we show for the Poisson equation with a right-hand side in L² that appropriate mesh refinement near some interior point diminishes the error at this point by nearly one order.