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

Author:      Susanne C. Brenner

Published:   Mathematics of Computation
             vol. 68 (1999), 559-583 (MR# 99i:65138)

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



Presented at:

- ILAY Workshop on Iterative Methods, 
  Toulouse, France, June 10-13, 1996

- Approximation and PDEs, Foundations of Computational Mathematics,
  Rio de Janeiro, Brazil, January 5--12, 1997

- Eighth Copper Mountain Conference on Multigrid Methods, 
  Copper Mountain, Colorado, April 6--11, 1997

- Third IMACS International Symposium on Iterative Methods 
  in Scientific  Computation, Jackson Hole, WY, July 9--12, 1997

- Guangzhou International Symposium on Computational Mathematics,
  Guangzhou, PRC, August 11-15, 1997 

- Innovative Finite Element Computations in Continuum Mechanics,
  15th IMACS World Congress, Berlin, Germany, August 24--29, 1997

- Imperial College, University of London, July 14, 1998

- University of Leeds, July 16, 1998

- University of Loughborough, July 17, 1998

- Schnelle Löser für partielle Differentialgleichungen, Oberwolfach,
  Germany, May 30--June 5, 1999





 paper: Mathematics of Computation