XXI Congreso de Ecuaciones Diferenciales y Aplicaciones XI Congreso de Matem´ atica Aplicada Ciudad Real, 21-25 septiembre 2009 (pp. 1–8) Multigrid finite element methods on semi-structured triangular grids F.J. Gaspar 1 , J.L. Gracia 1 , F.J. Lisbona 1 , C. Rodrigo 1 1 Applied Mathematics Department, University of Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain. E-mails: fjgaspar@unizar.es, jlgracia@unizar.es, lisbona@unizar.es, carmenr@unizar.es. Keywords: Geometric multigrid, Fourier analysis, triangular grids Abstract We are interested in the design of efficient geometric multigrid methods on hierar- chical 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 Fou- rier Analysis (LFA) predicts with high accuracy the multigrid convergence rates for different geometries. 1. Introduction Multigrid methods [3, 7, 8] are among the most efficient numerical algorithms for solving the large algebraic linear equation systems arising from discretizations of partial differential equations. In geometric multigrid, a hierarchy of grids must be proposed. For an irregular domain, it is very common to apply a refinement process to an unstructured input grid, such as Bank’s algorithm, used in the codes PLTMG [1] and KASKADE [5], obtaining a particular hierarchy of globally unstructured grids suitable for use with geometric multigrid. A simpler approach to generating the nested grids consists in carrying out several steps of repeated regular refinement, for example by dividing each triangle into four congruent triangles [4]. An important step in the analysis of PDE problems using finite element methods (FEM) is the construction of the large sparse matrix corresponding to the system of equa- tions to be solved. For discretizations of problems defined on structured grids with constant 1