Multigrid finite element methods on semi-structured triangular grids for planar elasticity

We are interested in the design of efficient geometric multigrid methods on hierarchical triangular grids for problems in two dimensions. Fourier analysis is a well-known useful tool in multigrid for the prediction of two-grid convergence rates which has been used mainly for rectangular–grids. This analysis can be extended straightforwardly to triangular grids by using an appropriate expression of the Fourier transform in a new coordinate systems, both in space and frequency variables. With the help of the Fourier Analysis, efficient geometric multigrid methods for the Laplace problem on hierarchical triangular grids are designed. Numerical results show that the Local Fourier Analysis (LFA) predicts with high accuracy the multigrid convergence rates for different geometries.