Meshfree Methods for Partial Differential Equations: The Particle-Partition of Unity Method

Meshfree methods have enjoyed a significant research effort in the past 15 years and substantial improvements have been accomplished and many different numerical schemes have been proposed. For instance, Smoothed Particle Hydrodynamics (SPH), Generalized Finite Difference Method (GFDM), Reproducing Particle Kernel Method (RKPM), Element Free Galerkin Method (EFGM), Generalized/eXtended Finite Element Methods (G/XFEM), or Partition of Unity Methods (PUM). The underlying construction principles in many of these scheme however are very similar and stem from scattered data approximation. In this talk we review these construction principles and discuss the approximation properties of the resulting meshfree function spaces.

In particular, we present the Particle-Partition of Unity Method (PPUM) which is a meshfree generalization of the classical finite element method. The PPUM can be employed in an h-version, a p-version and an hp-version. Furthermore, the PPUM supports the use of problem-dependent approximation spaces (i.e. there is a PPUM q-version) and it can be interpreted as a variational multi-scale method. We focus on hp-adaptive refinement of a PPUM discretization, the automatic hierarchical enrichment, and the efficient multilevel solution of the arising linear system. We present numerical results of our multilevel PPUM for the treatment of linear elastic fracture mechanics problems which demonstrate the approximation properties as well as the computational efficiency of the proposed scheme.