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BIT Numerical Mathematics

, Volume 37, Issue 3, pp 623–643 | Cite as

Multigrid methods for the computation of singular solutions and stress intensity factors II: Crack singularities

  • S. C. Brenner
  • L. -Y. Sung
Article

Abstract

We consider the Poisson equation −Δu=f with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain Ω 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 factors is\(\mathcal{O}(h^{(3/2) - \in } )\) whenfεL 2(Ω) and\(\mathcal{O}(h^{(2 - \in )} )\) whenfεH 1(Ω). The convergence rate in the energy norm is\(\mathcal{O}(h^{(1 - \in )} )\) in the first case and\(\mathcal{O}(h)\) in the second case. The costs of these multigrid methods are proportional to the number of elements in the triangulation. The general case wherefεH m (Ω) is also discussed.

AMS subject classification

65N55 65N30 

Key words

Multigrid crack singularities stress intensity factors 

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Copyright information

© BIT Foundation 1997

Authors and Affiliations

  • S. C. Brenner
    • 1
  • L. -Y. Sung
    • 1
  1. 1.Department of MathematicsUniversity of South CarolinaColumbia

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