Multigrid methods for the computation of singular solutions and stress intensity factors II: Crack singularities

We consider the Poisson equation −Δu=f with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain Ω with cracks. Multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed. The convergence rate for the stress intensity factors is whenfεL ²(Ω) and whenfεH ¹(Ω). The convergence rate in the energy norm is in the first case and in the second case. The costs of these multigrid methods are proportional to the number of elements in the triangulation. The general case wherefεH ^m (Ω) is also discussed. The work of the first author was partially supported by NSF under grant DMS-96-00133