Multicolor Fourier analysis of the multigrid method for quadratic FEM discretizations

Quadratic finite element methods offer some advantages for the numerical solution of partial differential equations (PDEs), due to their improved approximation properties in comparison to linear approaches. The algebraic linear systems arising from the discretization of PDEs by this kind of methods require an efficient resolution, and multigrid methods provide a good way to solve this problem. To design geometric multigrid methods, local Fourier analysis (LFA) is a very useful tool. However, LFA for quadratic finite element discretizations can not be performed in a standard way, since the discrete operator is defined by different stencils depending on the location of the points in the grid. In this work, a multicolor local Fourier analysis is presented to analyze multigrid solvers for this type of discretizations. With the help of this analysis, some point-wise and line-wise smoothers are analyzed. Some results showing the good correspondence between the two-grid convergence factors predicted by the analysis and the experimentally computed asymptotic convergence factors are presented. Finally, this analysis is applied to design a very efficient multigrid solver for semi-structured triangular grids, based on a hybrid-smoother.