Parallel 3 D Maxwell Solvers based on Domain Decomposition Data Distribution

The most efficient solvers for finite element (fe) equations are certainly multigrid, or multilevel methods, and domain decomposition methods using local multigrid solvers. Typically, the multigrid convergence rate is independent of the mesh size parameter, and the arithmetical complexity grows linearly with the number of unknowns. However, the standard multigrid algorithms fail for the Maxwell finite element equations in the sense that the convergence rate deteriorates as the mesh-size decreases. To overcome this drawback, R. Hiptmair proposed to modify the smoothing iteration by adding a smoothing step in the discrete potential space [Hip99]. Similarly, D. Arnold, R. Falk and R. Winther suggested a special block smoother that has the same effect [AFW00]. The parallelization of these or, more precisely, of appropriately modified multigrid solvers is certainly the only principle way to enhance the efficiency of these algorithms. Due to the peculiarities of the multigrid methods for the Maxwell equations, the parallelization is not straightforward. In this paper, we propose a unified approach to the parallelization of multigrid methods and domain decomposition methods. In order to develop a basic parallel Maxwell solver that can be used for more advanced problems as basic module, it is sufficient to consider the magnetostatic case. In the magnetostatic case, the Maxwell equation can be reduced to the curl-curl–equation that is not uniquely solvable because of the large kernel of the curl-operator (potential fields). In practice, a gauging condition is imposed in order to pick out a unique solution. The so-called Coulomb gauging aims at a divergence-free solution (vector potential). The weak formulation of the curl-curl–equation and the gauging condition together with a clever regularization leads to a regularized mixed variational formulation of the magnetostatic Maxwell equations in H0(curl) × H0(Ω) that has a unique solution. The discretization by the Nédélec and Lagrange finite elements results in a large, sparse, symmetric, but indefinite system of finite element equations. 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 regularizer in the regularization part and a special multigrid preconditioner for the regularized Nédélec finite element equations that we want to solve. (see second section). The parallelization of the