IEEE Transactions on Magnetics, Vol.34, No.5, Sept. 1998, pp. 3327-3330 13 An Algebraic Multigrid Method for Solving Very Large Electromagnetic Systems Ronny Mertens, Herbert De Gersem, Ronnie Belmans and Kay Hameyer Katholieke Universiteit Leuven, Dep. EE (ESAT), Div. ELEN, Kardinaal Mercierlaan 94, B-3001 Leuven, Belgium Domenico Lahaye, Stefan Vandewalle and Dirk Roose Katholieke Universiteit Leuven, Dep. Computer Science, Celestijnenlaan 200A, B-3001 Leuven, Belgium Abstract — Although most finite element programs have quite effective iterative solvers such as an incomplete Cholesky (IC) or symmetric successive overrelaxation (SSOR) preconditioned conjugate gradient (CG) method, the solution time may still become unacceptably long for very large systems. Convergence and thus total solution time can be shortened by using better preconditioners such as geometric multigrid methods. Algebraic multigrid methods have the supplementary advantage that no geometric information is needed and can thus be used as black box equation solvers. In case of a finite element solution of a non- linear magnetostatic problem, the algebraic multigrid method reduces the overall computation time by a factor of 6 compared to a SSOR-CG solver. Index terms — numerical analysis, electromagnetic analysis, iterative methods, finite element methods. I. INTRODUCTION In finite element programs, direct methods are nowadays often replaced by iterative methods to solve the system of discretized linear equations. Stationary methods such as Jacobi, Gauss-Seidel and successive overrelaxation are straightforward to implement but usually not very effective. The conjugate gradient method, a non-stationary method, is harder to apply, but very effective when used in combination with a good preconditioner. Symmetric successive overrelaxation and incomplete Cholesky decomposition are often used as preconditioners for the CG method. Multigrid methods can be used as solvers. Their use as preconditioner often results in even more efficient iterative methods. II. MULTIGRID METHODS The concept of multigrid methods is not new. The basic idea of this iterative method is to combine computed results obtained on different scales, using results from one scale to wipe out certain error components of the approximation of the solution on another scale (Fig. 1 & 2). An extensive treatment of multigrid methods can be found in [1]-[3]. Here, the main ideas are briefly recalled. The finite element discretization on a mesh of size h, of a diffusion problem, e.g. Poisson's equation, leads to a system of linear equations of the form A x b h h h = , (1) where A h is a sparse, symmetric and positive definite matrix, b h the right-hand side vector and x h the solution vector. Stationary iterative solvers have a general form of ( ) x x M A x b h k h k h h h k h + - = - - 1 1 , (2) e.g. Jacobi ( M h is the diagonal of A h ) or Gauss-Seidel ( M h is the lower triangular part of A h ). The error x x h h k - can be seen in the Fourier space as a linear combination of Fig. 1. Error after 0, 2 and 50 Jacobi iterations. Fig. 2. Error after 2 Jacobi iterations projected on a coarser grid. Manuscript received November 3, 1997. R. Mertens, +32 (0)16 321020, fax +32 (0)16 321985, Ronny.Mertens@esat.kuleuven.ac.be, http://www.esat.kuleuven.ac.be/ elen/elen.html; D. Lahaye, +32 (0) 16 327632, fax +32 (0)16 327996, Domenico.Lahaye@cs.kuleuven.ac.be, http://www.cs.kuleuven.ac.be/ cwis/dept-E.html. The authors are grateful to the Belgian "Fonds voor Wetenschappelijk Onderzoek Vlaanderen" for its financial support of this work and the Belgian Ministry of Scientific Research for granting the IUAP No. P4/20 on Coupled Problems in Electromagnetic Systems. The research Council of the K.U.Leuven supports the basic numerical research.