A Particle-Partition of Unity Method Part VIII: Hierarchical Enrichment

This paper is concerned with automatic enrichment in the particlepartition of unity method (PPUM). The goal of our automatic enrichment 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 automatically determined enrichment zone hierarchically near the singularities of the solution. To overcome the ill-conditioning of the enriched shape functions we present an appropriate local preconditioner. The results of our numerical experiments clearly show that the hierarchically enriched PPUM recovers the optimal convergence rate globally and even shows a kind of superconvergence within the enrichment zone. The condition number of the stiffness matrix is independent of the employed enrichment and the relative size of the enrichment zone.