Abstract
An efficient multigrid finite element method for vector problems on triangular anisotropic semi‐structured grids is proposed. This algorithm is based on zebra line‐type smoothers to overcome the difficulties arising when multigrid is applied on stretched meshes. In order to choose the type of multigrid cycle and the number of pre‐ and post‐smoothing steps, a three‐grid Fourier analysis is done. To this end, local Fourier analysis (LFA) on triangular grids for scalar problems is extended to the vector case. To illustrate the good performance of the method, a system of reaction‐diffusion is considered as model problem. A very satisfactory global convergence factor is obtained by using a V(0,2)‐cycle for domains triangulated with highly anisotropic meshes.
Highlights
One of the most important aspects in the numerical solution of systems of partial differential equations is the efficient solution of the corresponding large systems of equations arising from their discretization
Local Fourier analysis (LFA) on triangular grids for scalar problems is extended to the vector case
While algebraic multigrid is capable of handling large problems with irregular structure, geometric multigrid always has a lower cost per iteration, because of its ability to take advantage of the geometry within the data structures used
Summary
One of the most important aspects in the numerical solution of systems of partial differential equations is the efficient solution of the corresponding large systems of equations arising from their discretization. Exploiting the regularity of the grid, geometric multigrid methods can be implemented using stencil-based operations, that is, if finite element discretizations are considered it is not necessary to construct the global matrix by assembly, drastically reducing the memory required, see [2] Another interesting point is to use the LFA on triangular grids [5] to choose the good components of the multigrid method for each triangular input block, taking into account the particular geometry of the grid on each patch. This strategy has been performed for scalar problems, and its extension to vector problems is pointed out in this paper From this analysis, zebra line-type smoothers appear as a good relaxation method for these anisotropic semi-structured grids, providing a very efficient multigrid algorithm. It will be seen that a V(0,2) multigrid method seems to be a very good solver for the proposed problems
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