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

We consider the Poisson equation $-\Delta u=f$ with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain $\Omega$ with re-entrant angles. A multigrid method for the computation of singular solutions and stress intensity factors using piecewise linear functions is analyzed. When $f\in L^{2}(\Omega )$, the rate of convergence to the singular solution in the energy norm is shown to be ${\mathcal{O}}(h)$, and the rate of convergence to the stress intensity factors is shown to be ${\mathcal{O}}(h^{1+(\pi /\omega )-\epsilon })$, where $\omega$ is the largest re-entrant angle of the domain and $\epsilon >0$ can be arbitrarily small. The cost of the algorithm is ${\mathcal{O}}(h^{-2})$. When $f\in H^{1}(\Omega )$, the algorithm can be modified so that the convergence rate to the stress intensity factors is ${\mathcal{O}}(h^{2-\epsilon })$. In this case the maximum error of the multigrid solution over the vertices of the triangulation is shown to be ${\mathcal{O}}(h^{2-\epsilon })$.