A particle-partition of unity method part VI: A p-robust multilevel solver

Abstract

In this paper we focus on the efficient multilevel solution of linear systems arising from a higher order discretization of a second order partial differential equation using a partition of unity method. We present a multilevel solver which employs a tree-based spatial multilevel sequence in conjunction with a domain decomposition type smoothing scheme. The smoother is based on an overlapping subspace splitting, where the subspaces contain all interacting local polynomials. The resulting local subspace problems are solved exactly. This leads to a computational complexity of the order O(Np3d) per iteration. The results of our numerical experiments indicate that the convergence rate of this multilevel solver is independent of the number of points N and the approximation order p. Hence, the overall complexity of the solver is of the order O(log(1/ε)Np3d) to reduce the initial error by a prescribed factor ε.