The authors study local refinements for boundary value problems with singularities in the framework of multigrid and finite-difference methods. The model Poisson problem with a-priori defined refinements is used. The conclusions of the paper are supported by heuristic arguments and computational experiments. The method uses rectangular grid patches for refinement with the multigrid FAS scheme. Local relaxations can restore multigrid convergence rates to the values that would be attained in the absence of singularities. An ''exchange-rate'' algorithm is introduced to maintain linear dependence between computational work and number of gridpoints. The grading of the mesh is governed by an ordinary differential equation for optimizing the step size, derived from an assumption that the error of the solution can be approximated as a weighted integral of the magnitude of the truncation error. The results of numerical experiments show that no refinement is needed in the case of a point source singularity for regular accuracy far from the source. A technique is presented for retaining conservation form with local refinement.