A Multigrid Method for the Poisson Equation with Corner Singularities

Zhiqiang Cai

Department of Mathematics Purdue University 1395 Mathematical Sciences Building West Lafayette, IN 47907-1395


Abstract

In this talk, we first introduce a new finite element method for the Poisson equation with homogeneous Dirichlet boundary conditions on a polygonal domain with one re-entrant angle. It is well-known that the solution of such problem has a singular representation: $u=w+\lambda \eta s$ where $w\in H^2(\O)\cap H^1_0(\O)$, $\lambda\in {\cal R}$ and $\eta$ are the respective stress intensity factor and cut-off function, and $s$ is a known singular function depending only on the re-entrant angle. By using the dual singular and an extra cut-off functions, we are able to deduce a well-posed variational problem for $w$ and an extraction formula for $\lambda$ in terms of $w$. Standard continuous piecewise linear finite element approximation yields $O(h)$ optimal accuracy for $w$, which, in turn, implies the same accuracy for $u$, in the $H^1$ norm. We are only able to prove $O(h^{1+\frac\pi\o -\e})$ error bounds for $w$ and $u$ in the $L^2$ norm and for $\lambda$ in the absolute value, where $\o$ is the internal angle and $\epsilon$ is any positive number. But numerical experiments for a particular model problem suggest that our method achieves $O(h^{2-\epsilon})$ accuracy. Multigrid $V(1,0)$-cycle applied to the algebraic equations resulting from the finite element discretization of $w$ is shown to be convergent independent of the finest mesh size $h$ and the number of levels, provided that the coarsest mesh size is sufficient small. Numerical experiments on the convergence rate of multigrid algorithms are also provided.