Abstract: Mathematical models describing physics and engineering problems often require the resolution of partial differential equations. To this end, discretizations based on finite difference, finite volumes or finite elements are considered to adequately approximate the continuous problem, properly capturing and retaining the characteristics of the underlying physical processes. Since often these models must be solved on complex domains, it is popular to discretize the problem on an unstructured partitioning of the domain.
The large sparse algebraic linear systems arising from the discretization of PDEs require an efficient resolution, and multigrid algorithms are among the most efficient methods for solving this kind of systems. Algebraic multigrid methods appear as the most suitable to handle unstructured grids. However, our interest lies in the combination of geometric multigrid methods, which take advantage of the regularity of structured meshes, with the use of semi-structured grids. This type of meshes combines the flexibility of a totally unstructured input grid to capture the geometry of the domain, with the advantages obtained from the structured patches arising from the regular refinement of the initial triangular elements, where the implementation of a geometric multigrid method can be done very efficiently.
Since the good performance of the method depends on the particular choice of the components of the algorithm for an individual problem, the local Fourier analysis (LFA) is often used to predict the convergence rates of the multigrid method, and to design suitable components. In particular, LFA on triangular grids is used in the framework of semi-structured grids, being applied to each triangular block of the initial unstructured grid to choose suitable local components giving rise to a block-wise multigrid algorithm which becomes a very efficient solver.