Title:      Multigrid method for the computation of singular solutions 
            and stress intensity factors. II: Crack singularities
 
Authors:    Susanne C. Brenner and Li-yeng Sung

Published:  BIT, v. 37 (1997), 623-643 (MR# 99i:65139)

Abstract:   We consider the Poisson equation $-\Delta u=f$ with homogeneous 
            Dirichlet boundary condition on a two-dimensional polygonal 
            domain $\Omega$ with cracks. Multigrid methods for the computation 
            of singular solutions and stress intensity factors using piecewise 
            linear functions are analyzed. The convergence rate for the stress 
            intensity factorsis ${\cal O}(h^{(3/2)-\epsilon})$ when 
            $f\in L^2(\Omega)$ and ${\cal O}(h^{2-\epsilon})$ when 
            $f\in H^1(\Omega)$.  The convergence rate in the energy norm is 
            ${\cal O}(h^{1-\epsilon})$ in the first case and ${\cal O}(h)$ in 
            the second case.  The costs of these multigrid methods are  
            proportional to the number of elements in the triangulation.  The 
            general case where $f\in H^m(\Omega)$ is also discussed.  



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