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In this paper a local Fourier analysis for multigrid methods on tetrahedral grids is presented. Different smoothers for the discretization of the Laplace operator by linear finite elements on such grids are analyzed. A four-color smoother is presented as an efficient choice for regular tetrahedral grids, whereas line and plane relaxations are needed for poorly shaped tetrahedra. A novel partitioning of the Fourier space is proposed to analyze the four-color smoother. Numerical test calculations validate the theoretical predictions. A multigrid method is constructed in a block-wise form, by using different smoothers and different numbers of pre-and post-smoothing steps in each tetrahedron of the coarsest grid of the domain. Some numerical experiments are presented to illustrate the efficiency of this multigrid algorithm.

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SUMMARYMultigrid (MG) methods are among the most efficient and widespread methods for solving large linear systems of equations that arise, for example, from the discretization of partial differential equations. In this paper we introduce a new approach for optimizing the computational cost of the full MG method to achieve a given accuracy by determining the number of MG cycles on each level. To achieve this, a very efficient and flexible Branch and Bound algorithm is developed. The implementation in the parallel finite element solver Hierarchical Hybrid Grids leads to a significant reduction in CPU time.

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