CENTER FOR COMPUTATIONAL MATHEMATICS COLLOQUIUM

                  UNIVERSITY OF COLORADO AT DENVER



TITLE:   Multigrid Methods for Stress Intensity Factors and Singular Solutions
 

SPEAKER: Susanne C.  Brenner, Department of Mathematics, 
         University of South Carolina, Columbia, SC 
         

DATE:    Monday, November 4, 1996

PLACE:   Math Conference Room - Suite 540 
         UCD Building, 1250 14th St., Denver

TIME:    11 am (Refreshments served at 10:45 am) (NOTE UNUSUAL TIME)



ABSTRACT


(for a post-script version of the abstract press here)



Let $\O$ be a bounded polygonal domain (possibly with cracks) in $\R^2$.  
Consider the Poisson equation with homogeneous Dirichlet boundary condition:
     $$\eqalign{
     -\Delta u &= f \quad \hbox{in}\quad \O \cr        
           u &= 0 \quad \hbox{on}\quad\partial\O\cr}$$
where $f \in L^2(\O)$.
\par
Let $\omega_1, \ldots, \omega_J$ be the re-entrant angles of $\O$ and $p_j$ be the corresponding vertices.  It is well-known  that the unique
solution $u \in H^1_0(\O)$ has the representation
     $$ u = \sum_{j=1}^J \kappa_j s_j + w,$$
where $w\in H^{2-\epsilon}(\O)$ and $s_j\in H^{1+(\pi/\omega_j)-\epsilon}(\O)$.
\par
In this talk we present a multigrid method for the computation of $w$ and the
stress intensity factors $\kappa_j$ using linear finite elements
on quasi-uniform grids.  The resulting approximate solution has the form
$u_h=\sum_{j=1}^J \kappa_{j,h}s_j + w_h$, where $w_h$ is a piecewise linear
function.  The convergence rate of $w_h$ to $w$ in the energy norm is  
${\cal O}(h^{1-\epsilon})$, and 
the convergence rate of $\kappa_{j,h}$  to $\kappa_j$ is  
${\cal O}(h^{1+(\pi/\omega)-\epsilon})$, where
$\omega$ is the largest re-entrant angle of the domain.
This method can be modified to produce 
${\cal O}(h^{2-\epsilon})$ convergence for the stress intensity factors when 
the function $f\in H^1(\O)$.
\par
We will also discuss the general case where $f\in H^m(\O)$ for $m\geq2$.
Using the Lagrange ${\rm P}_{m+1}$ element the stress intensity factors
can be computed at an ${\cal O}(h^{m+1-\epsilon})$ convergence rate by a 
multigrid method on quasi-uniform grids.
\par
The costs of all the algorithms are proportional to the number of elements in
the triangulation.




(for a post-script version of the abstract press here)