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Multilevel particle-partition of unity method. (English) Zbl 1218.65132

Summary: This paper is concerned with the particle-partition of unity method, a meshfree generalization of the finite element method. We present the fundamental construction principles and abstract approximation properties of the resulting function spaces \(V^{\text{PU}}\). Moreover, we discuss the construction of optimal approximation spaces for a reference application in linear elastic fracture mechanics in particular. The presented construction not only yields optimal convergence rates globally independently of the regularity of the solution, our method shows a super-convergence near the singular points of the solution.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
74B05 Classical linear elasticity
74R10 Brittle fracture
74S05 Finite element methods applied to problems in solid mechanics

Software:

PUMA
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Full Text: DOI

References:

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