AND SINGULAR SOLUTIONS

Department of Mathematics

University of South Carolina

Columbia, SC 29208

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 L^{2}.
It is well-known that the unique solution u in the Sobolev space
H^{1} has a singular function representation
u=(k_{1}s_{1}+k_{2}s_{2}
+...+k_{J}s_{J})+w, where s_{j} is the singular
function associated with the j^{th} re-entrant angle,
the k_{j}'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 H^{2}, 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 H^{1}.

We will also discuss the general case where the right-hand side is in
the Sobolev space H^{m}. Using the Lagrange P_{m+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.