Convergence analysis of a hierarchical enrichment of Dirichlet boundary conditions in a mesh-free method

Implementation of Dirichlet boundary conditions in mesh-free methods is problematic. In Wagner and Liu (International Journal for Numerical Methods in Engineering 2001; 50:507), a hierarchical enrichment technique is introduced 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 enrichment 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 ID and 2D are included showing the convergence behaviour of mesh-free interpolants and mesh-free solutions when the hierarchical enrichment mesh-free technique is employed.