#### Title

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

#### Document Type

Article

#### Publication Date

1-1-1999

#### Abstract

We consider the Poisson equation -Δu = f with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain Ω 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 ∈ L (Ω), the rate of convergence to the singular solution in the energy norm is shown to be Script O sign(h), and the rate of convergence to the stress intensity factors is shown to be Script O sign(h , where w is the largest re-entrant angle of the domain and ∈ > 0 can be arbitrarily small. The cost of the algorithm is Script O sign(h ). When f ∈ H (Ω), the algorithm can be modified so that the convergence rate to the stress intensity factors is Script O sign(h ). In this case the maximum error of the multigrid solution over the vertices of the triangulation is shown to be Script O sign(h ). 2 1+(π/w)-∈) -2 1 2-∈ 2-∈

#### Publication Source (Journal or Book title)

Mathematics of Computation

#### First Page

559

#### Last Page

583

#### Recommended Citation

Brenner, S.
(1999). Multigrid methods for the computation of singular solutions and stress intensity factors I: Corner singularities.* Mathematics of Computation**, 68* (226), 559-583.
https://doi.org/10.1090/s0025-5718-99-01017-0