The hierarchical preconditioning on unstructured three-dimensional grids with locally refined regions

This paper presents two hierarchically preconditioned methods for the fast solution of mesh equations that approximate three-dimensional-elliptic boundary value problems on unstructured quasiuniform triangulations above all aiming at the numerical investigation of the previously suggested algorithms. Furthermore, improving the practical applicability of the methods unstructured three-dimensional grids possessing locally refined regions are considered. Based on the fictitious space approach, the original problem can be adaptively embedded into an auxiliary one in which hanging nodes occur. We implemented the corresponding Yserentant preconditioned conjugate gradient method as well as the BPX-preconditioned cg-iteration having nearly optimal computational costs. Several numerical examples demonstrate the efficiency of the artificially constructed hierarchical methods.