A survey of parallel solvers for the finite element method in computational electromagnetics <s> Multilevel methods <s> The main purpose of this paper is to give a systematic introduction to a number of iterative methods for symmetric positive definite problems. Based on results and ideas from various existing works on iterative methods, a unified theory for a diverse group of iterative algorithms, such as Jacobi and Gauss–Seidel iterations, diagonal preconditioning, domain decomposition methods, multigrid methods, multilevel nodal basis preconditioners and hierarchical basis methods, is presented. By using the notions of space decomposition and subspace correction, all these algorithms are classified into two groups, namely parallel subspace correction (PSC) and successive subspace correction (SSC) methods. These two types of algorithms are similar in nature to the familiar Jacobi and Gauss–Seidel methods, respectively.A feature of this framework is that a quite general abstract convergence theory can be established. In order to apply the abstract theory to a particular problem, it is only necessary to specify a decomposi... <s> BIB001 </s> A survey of parallel solvers for the finite element method in computational electromagnetics <s> Multilevel methods <s> Abstract Three-dimensional magnetic field problems are challenging not only because of interesting applications in the industry but also from the mathematical point of view. In the magnetostatic case, our Maxwell solver is based on a regularized mixed variational formulation of the Maxwell equations in H 0 ( curl )×H 0 1 (Ω) and their discretization by the Nedelec and Lagrange finite elements. Eliminating the Lagrange multiplier from the mixed finite element equations, we arrive at a symmetric and positive definite (spd) problem that can be solved by some parallel multigrid preconditioned conjugate gradient (pcg) method. More precisely, this pcg solver contains a standard scaled Laplace multigrid regularizator in the regularization part and a special multigrid preconditioner for the regularized Nedelec finite element equations that we want to solve. The parallelization of the pcg algorithm, the Laplace multigrid regularizator and the multigrid preconditioner are based on a unified domain decomposition data distribution concept. We present the results of some numerical experiments on a parallel machine with distributed memory that show the high efficiency of our approach to real-life applications. <s> BIB002
Multilevel methods can be included in the class of decomposition method, since these algorithms have similarities with Schwarz type domain decomposition method with inexact solvers. Different grid levels or subspaces play the role of subregions. Schematically, each iteration may be seen as performed into 3 steps ( fig. 11) . A few iterations are first performed on a fine grid, leading to an approximative solution u h . This leads to a residual vector r h which is transferred to the coarse grid and then exactly computed (second step). The solution is finally re-transferred onto the fine grid, allowing to modify the previous values of u h . It is followed by a post-smoothing step. Multilevel methods can be generalized to several grid sizes. Two types of multigrid methods exist  : -the Geometric Multigrid Method (GMM), based on the effective use of mesh grids, -the Algebraic Multigrid Method (AMG), where there is no mesh, and the building of the operators is based on the values of the coefficients of the matrix. Parallel implementation of the AMG is presented in  for eddy-current analyses and magnetostatics analyses. The parallel performance of the algorithm is improved in the setup phase by the use of longrange interpolation instead of the conventional direct interpolation. The numerical results obtained with a rectangular domain show that the AMG is a fast solver, since the CPU time is divided by 10 compared to a classical ICCG (Table III) . As shown in Fig. 12 , this intrisic property is not destroyed by the parallelization of the method, since both speedups remain comparable. It would be of great interest to test the AMG method with more complicated domains, since the GMG requires a hierarchy of coarse grids which is actually difficult to manage in 3D. An experience of implementation of the GMG has been reported in BIB002 for 3D magnetostatics. Note that all these methods (domain decomposition or multigrid methods) can be used as preconditioners for the Conjugate Gradient algorithm BIB001 . 