Stable enrichment and treatment of complex domains in the particle–partition of unity method
Wednesday, November 3, 2010 - 12:00pm - 12:45pm
We are concerned with the stability and approximation properties of enriched meshfree methods for the discretization of PDE on arbitrary domains. In particular we focus on the particle-partition of unity method (PPUM) yet the presented results hold for any partition of unity based enrichment scheme. The goal of our enrichment scheme is to recover the optimal convergence rate of the uniform h-version independent of the regularity of the solution. Hence, we employ enrichment not only for modeling purposes but rather to improve the approximation properties of the numerical scheme. To this end we enrich our PPUM function space in an enrichment zone hierarchically near the singularities of the solution. This initial enrichment however can lead to a severe ill-conditioning and can compromise the stability of the discretization. To overcome the ill-conditioning of the enriched shape functions we present an appropriate local preconditioner which yields a stable basis independent of the employed initial enrichment. The construction of this preconditioner is of linear complexity with respect to the number of discretization points. The treatment of general domains with mesh-based methods such as the finite element method is rather involved due to the necessary mesh-generation. In collocation type meshfree methods this complex pre-processing step is completely avoided by construction. However, in Galerkin type meshfree discretization schemes we must compute domain and boundary integrals and thus must be concerned with the meshfree treatment of arbitrary domains. Here, we present a cut-cell-type scheme for the partition of unity method and ensure stability by enforcing the flat-top condition in a simple post-processing step.