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Convergence analysis of a hierarchical enrichment of Dirichlet boundary conditions in a mesh-free method. (English) Zbl 0995.65108
Summary: Implementation of Dirichlet boundary conditions in mesh-free methods is problematic. G. J. Wagner and W. K. Liu [Int. Numer. Methods Eng. 50, No. 3, 507–524 (2001; Zbl 1006.76073)] introduced, a hierarchical enrichment technique that allows a simple implementation of the Dirichlet boundary conditions. In this paper, we provide some error analysis for the hierarchical enrichment mesh-free technique. We derive optimal order error estimates for the hierarchical enrichmet mesh-free interpolants. For one-dimensional elliptic boundary value problems, we can directly apply the interpolation error estimates to obtain error estimates for the mesh-free solutions. For higher-dimensional problems, derivation of error estimates for the mesh-free solutions depends on the availability of an inverse inequality. Numerical examples in one-dimensional and two-dimensional are included showing the convergence behaviour of mesh-free interpolants and mesh-free solutions when the hierarchical enrichment mesh-free technique is employed.

MSC:
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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