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Multigrid methods for the computation of singular solutions and stress intensity factors. II: Crack singularities. (English) Zbl 0890.73060

Summary: 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 factors is \({\mathcal O} (h^{(3/2) -\varepsilon})\) when \(f\in L^2 (\Omega)\), and \({\mathcal O} (h^{2-\varepsilon})\) when \(f\in H^1 (\Omega)\). The convergence rate in the energy norm is \({\mathcal O} (h^{1-\varepsilon})\) 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 where \(f\in H^m (\Omega)\) is also discussed.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74R99 Fracture and damage
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74G70 Stress concentrations, singularities in solid mechanics
74H35 Singularities, blow-up, stress concentrations for dynamical problems in solid mechanics
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