On geometric multigrid methods for triangular grids using three-coarsening strategy

Abstract

This paper deals with the design of efficient geometric multigrid methods on hierarchical triangular grids using a three-coarsening strategy. In [F.J. Gaspar, J.L. Gracia, F.J. Lisbona, Fourier analysis for multigrid methods on triangular grids, SIAM J. Sci. Comput., in press], a Local Fourier Analysis (LFA) for multigrid methods with standard coarsening on triangular grids has been proposed. It is based on an expression of the Fourier transform in new coordinate systems. LFA is applied to the new coarsening strategy to design components for an efficient multigrid algorithm. The definition of low and high frequencies, and therefore the spaces of harmonics, are adapted according to the new situation. Appropriate smoothing methods, in particular a three-color smoother with optimal relaxation parameter, are proposed and analyzed for the discrete Laplace operator obtained with linear finite elements. Moreover, special inter-grid transfer operators are designed. These methods are compared with standard coarsening algorithms in terms of the computational work required. Independently of the shape of the triangles, we show that the three-coarsening strategy is a good computational alternative to standard coarsening.