An improved multigrid solver for the p-hierarchical basis finite element method using a space decomposition smoother

Abstract

An improved hierarchical basis finite element multigrid solver for the p -version is presented and analysed. The proposed solver combines the features of p -hierarchical bases, multigrid and space decomposition effectively. Unlike the existing p -version multigrid solvers that perform smoothing only on a subspace of the multigrid space, we adopt a global smoothing strategy at each multigrid level via space decomposition, in a multiplicative Schwarz manner. The finite element subspaces are constructed on locally uniform p -refinements, yielding efficient implementations on p -hierarchical bases. Moreover, the error operator has been analysed and the p -version strengthened Cauchy Buniakowskii Schwarz inequality constants have been established yielding the first convergence estimates. Furthermore, numerical examples are given confirming the effectiveness of the solver. The numerical experiments show that the proposed solver outperforms the benchmark p -version space decomposition solver in terms of the convergence rates as well as the computational complexity.