Introduction to Multigrid Methods for Elliptic Boundary Value Problems Arnold Reusken Institut f¨ ur Geometrie und Praktische Mathematik RWTH Aachen, D-52056 Aachen, Germany E-mail: reusken@igpm.rwth-aachen.de. We treat multigrid methods for the efficient iterative solution of discretized elliptic boundary value problems. Two model problems are the Poisson equation and the Stokes problem. For the discretization we use standard finite element spaces. After discretization one obtains a large sparse linear system of equations. We explain multigrid methods for the solution of these linear systems. The basic concepts underlying multigrid solvers are discussed. Results of numerical experiments are presented which demonstrate the efficiency of these method. Theoretical con- vergence analyses are given that prove the typical grid independent convergence of multigrid methods. 1 Introduction In these lecture notes we treat multigrid methods (MGM) for solving discrete elliptic boundary value problems. We assume that the reader is familiar with discretization meth- ods for such partial differential equations. In our presentation we apply on finite element discretizations. We consider the following two model problems. Firstly, the Poisson equa- tion −Δu = f in Ω ⊂ R d , u =0 on ∂ Ω, (1) with f a (sufficiently smooth) source term and d =2 or 3. The unknown is a scalar function u (for example, a temperature distribution) on Ω. We assume that the domain Ω is open, bounded and connected. The second problem consists of the Stokes equations −Δu + ∇p = f in Ω ⊂ R d , div u =0 in Ω, u =0 on ∂ Ω. (2) The unknowns are the velocity vector function u =(u 1 ,...,u d ) and the scalar pressure function p. To make this problem well-posed one needs an additional condition on p, for example,  Ω p dx =0. Both problems belong to the class of elliptic boundary value prob- lems. Discretization of such partial differential equations using a finite difference, finite volume or finite element technique results in a large sparse linear system of equations. In the past three decades the development of efficient iterative solvers for such systems of equations has been an important research topic in numerical analysis and computational en- gineering. Nowadays it is recognized that multigrid iterative solvers are highly efficient for this type of problems and often have “optimal” complexity. There is an extensive literature on this subject. For a thorough treatment of multigrid methods we refer to the monograph 1