Fourier Analysis for Multigrid Methods on Triangular Grids

In this paper a local Fourier analysis technique for multigrid methods on triangular grids is presented. The analysis is based on an expression of the Fourier transform in new coordinate systems, both in space variables and in frequency variables, associated with reciprocal bases. This tool makes it possible to study different components of the multigrid method in a very similar way to that of rectangular grids. Different smoothers for the discrete Laplace operator obtained with linear finite elements are analyzed. A new three-color smoother has been studied and has proven to be the best choice for “near” equilateral triangles. It is also shown that the block-line smoothers are more appropriate for irregular triangles. Numerical test calculations validate the theoretical predictions.