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The purpose of these notes is to present, at graduate level, an introduction to the application of multigrid methods to elliptic and hyperbolic partial dierential equations for engineers, physicists and applied mathematicians. The reader is assumed to be familiar with the basics of the analysis of partial dierential equations and of numerical mathematics, but the use of more advanced mathematical tools, such as functional analysis, is avoided. The course is intended to be accessible to a wide audience of users of computational methods. We do not, therefore, delve deeply into the mathematical foundations. This is done in the excellent monograph by Hackbusch , which treats many aspects of multigrid, and also contains many practical details. The book  is more accessible to non-mathematicians, and pays more attention to applications, especially in computational Fuid dynamics.