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Mathematics of Computation

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

Author: Susanne C. Brenner
Journal: Math. Comp. 68 (1999), 559-583
MSC (1991): Primary 65N55, 65N30
MathSciNet review: 1609670
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Abstract: We consider the Poisson equation $-\Delta u=f$ with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain $\Omega $ with re-entrant angles. A multigrid method for the computation of singular solutions and stress intensity factors using piecewise linear functions is analyzed. When $f\in L^{2}(\Omega )$, the rate of convergence to the singular solution in the energy norm is shown to be ${\mathcal{O}}(h)$, and the rate of convergence to the stress intensity factors is shown to be ${\mathcal{O}}(h^{1+(\pi /\omega )-\epsilon })$, where $\omega $ is the largest re-entrant angle of the domain and $\epsilon >0$ can be arbitrarily small. The cost of the algorithm is ${\mathcal{O}}(h^{-2})$. When $f\in H^{1}(\Omega )$, the algorithm can be modified so that the convergence rate to the stress intensity factors is ${\mathcal{O}}(h^{2-\epsilon })$. In this case the maximum error of the multigrid solution over the vertices of the triangulation is shown to be ${\mathcal{O}}(h^{2-\epsilon })$.

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Additional Information

Susanne C. Brenner
Affiliation: Department of Mathematics, University of South Carolina, Columbia, SC 29208

Keywords: Multigrid, corner singularities, stress intensity factors, superconver\-gence
Received by editor(s): July 2, 1996
Additional Notes: This work was supported in part by the National Science Foundation under Grant Nos. DMS-94-96275 and DMS-96-00133.
Article copyright: © Copyright 1999 American Mathematical Society