We consider the Poisson equation with homogeneous Dirichlet boundary condition on a bounded two-dimensional polygonal domain with re-entrant angles (and possibly cracks) where the right-hand side is in L2. It is well-known that the unique solution u in the Sobolev space H1 has a singular function representation u=(k1s1+k2s2 +...+kJsJ)+w, where sj is the singular function associated with the jth re-entrant angle, the kj's are the stress intensity factors, and w is the regular part. Because of the singular functions, the solution u is singular in the sense that it is not in H2, and the convergence rates of standard finite element methods are adversely affected.
In this talk we will show that the convergence rates can be improved using standard finite elements on quasi-uniform grids by combining the full multigrid nested iteration technique and the singular function representation. We will present a multigrid method for the computation of the singular solution u and the stress intensity factors using linear finite elements on quasi-uniform grids. The resulting approximate solution to u is a linear combination of the singular functions and a piecewise linear function. The convergence rate of the approximation to u is of order almost 1 in the energy norm, and the convergence rate of the approximate stress intensity factors is of order almost 1 + pi/a, where a is the measure of the largest re-entrant angle. This method can be modified to produce an almost order 2 convergence rate for the stress intensity factors when the right-hand side is in the Sobolev space H1.
We will also discuss the general case where the right-hand side is in the Sobolev space Hm. Using the Lagrange Pm+1 element, the stress intensity factors can be computed with a convergence rate of order almost m+1 by a multigrid method on quasi-uniform grids.
The costs of all the algorithms are proportional to the number of elements in the triangulation.