Techniques are considered for improving the numerical approximation to a problem which possesses corner singularities. The particular problem considered here is taken from solid state electronics. Relined numerical solutions are obtained using two techniques. The first employs a non-uniform grid with a smaller spacing of the mesh points in the neighbourhood of the singularities. The second technique involves a transformation of the problem to one which is free from singularities. The dynamic ADI method, which needs no a priori relaxation parameters to accelerate convergence, is used to solve the discretized problem in both cases.