Stable enrichment and local preconditioning in the particle-partition of unity method

This paper is concerned with the stability and approximation properties of enriched meshfree methods. 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 and optimally conditioned basis independent of the employed initial enrichment. The construction of this preconditioner is of linear complexity with respect to the number of discretization points. We obtain optimal error bounds for an enriched PPUM discretization with local preconditioning that are independent of the regularity of the solution globally and within the employed enrichment zone we attain a kind of super-convergence. The results of our numerical experiments clearly show that our enriched PPUM with local preconditioning recovers the optimal convergence rate of O(h) of the uniform h-version globally. For the considered model problems from linear elastic fracture mechanics we obtain an improved convergence rate of O(h) with δ ≥ 1 2 for p = 1. The condition number of the stiffness matrix is independent of the employed enrichment zone.