A Particle-Partition of Unity Method Part VII: Adaptivity

This paper is concerned with the adaptive multilevel solution of elliptic partial differential equations using the partition of unity method. While much of the work on meshfree methods is concerned with convergence-studies, the issues of fast solution techniques for the discrete system of equations and the construction of optimal order algorithms are rarely addressed. However, the treatment of large scale real-world problems by meshfree techniques will become feasible only with the availability of fast adaptive solvers. The adaptive multilevel solver proposed in this paper is a main step toward this goal. In particular, we present an h-adaptive multilevel solver for the partition of unity method which employs a subdomain-type error indicator to control the refinement and an efficient multilevel solver within a nested iteration approach. The results of our numerical experiments in two and three space dimensions clearly show the efficiency of the proposed scheme.