# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## Multigrid methods for the computation of singular solutions and stress intensity factors I: Corner singularitiesHTML articles powered by AMS MathViewer

by Susanne C. Brenner
Math. Comp. 68 (1999), 559-583 Request permission

## 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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