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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. First published online: 09 Jun 2011