It is well-known that solutions of two-dimensional elliptic boundary value problems suffer a loss of regularity near a point where there is a geometric singularity (e.g., a reentrant corner or a crack tip), an abrupt change in the boundary conditions (e.g., Dirichlet/Neumann), or a discontinuity of the coefficients of the differential operator (e.g., a cross point of an interface problem). This loss of regularity is detrimental to the convergence of standard finite element methods using low order elements and quasi-uniform grids. Traditonal remedies include the p-version of the finite element method and local mesh-refinement.

In this talk we show that using a multilevel approach the optimal convergence rate for the P1 finite element method on quasi-uniform grids can be recovered at the presence of singular points, provided we combine the full multigrid methodology with the singular function representation of the solution of the boundary value problem and the extraction formulas that express the stress intensity factors in terms of the solution.