Multigrid methods for the computation of singular solutions and stress intensity factors I: Corner singularities

We consider the Poisson equation −∆u = f with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain Ω with re-entrant angles. A multigrid method for the computation of singular solutions and stress intensity factors using piecewise linear functions is analyzed. When f ∈ L2(Ω), the rate of convergence to the singular solution in the energy norm is shown to be O(h), and the rate of convergence to the stress intensity factors is shown to be O(h1+(π/ω)− ), where ω is the largest re-entrant angle of the domain and > 0 can be arbitrarily small. The cost of the algorithm is O(h−2). When f ∈ H1(Ω), the algorithm can be modified so that the convergence rate to the stress intensity factors is O(h2− ). In this case the maximum error of the multigrid solution over the vertices of the triangulation is shown to be O(h2− ).