A survey of parallel solvers for the finite element method in computational electromagnetics <s> Non overlapping domain methods <s> A novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented. The spatial domain is partitioned into a set of totally disconnected subdomains, each assigned to an individual processor. Lagrange multipliers are introduced to enforce compatibility at the interface nodes. In the static case, each floating subdomain induces a local singularity that is resolved in two phases. First, the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution. Next, the contributions of these modes are related to the Lagrange multipliers through an orthogonality condition. A parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and Lagrange multipliers, which completes the solution of the problem. When implemented on local memory multiprocessors, this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring. It is also suitable for parallel/vector computers with shared memory. Moreover, unlike parallel direct solvers, it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations. <s> BIB001 </s> A survey of parallel solvers for the finite element method in computational electromagnetics <s> Non overlapping domain methods <s> Abstract An efficient domain decomposition (DD) algorithm for solving nonlinear boundary value problems on multiple instruction multiple data parallel computers is presented. The linearization is done by Newton's method, and the linear system is solved by CG with DD preconditioning. Further, the DD approach allows finite element and Galerkin boundary element methods to be combined in a unifled variational problem. Test computations on a transputer-based system were performed for both model problems and a real-life electric motor. <s> BIB002 </s> A survey of parallel solvers for the finite element method in computational electromagnetics <s> Non overlapping domain methods <s> We present a new finite element (FE) method for magnetotelluric modelling of three-dimensional conductivity structures. Maxwell's equations are treated as a system of first-order partial differential equations for the secondary fields. Absorbing boundary conditions are introduced, minimizing undesired boundary effects and allowing the use of small computational domains. The numerical algorithm presented here is an iterative, domain decomposition procedure employing a nonconforming FE space. It does not use global matrices, therefore allowing the modellization of large and complicated structures. The algorithm is naturally parallellizable, and we show results obtained in the IBM SP2 parallel supercomputer at Purdue University. The accuracy of the numerical method is verified by checking the computed solutions with the results of COMMEMI, the international project on the comparison of modelling methods for electromagnetic induction. <s> BIB003 </s> A survey of parallel solvers for the finite element method in computational electromagnetics <s> Non overlapping domain methods <s> A domain decomposition method based on the finite-element tearing and interconnecting (FETI) algorithm is presented for the solution of the large sparse matrices associated with the finite-element method (FEM) solution of the vector wave equation. The FETI algorithm is based on the method of Lagrange multipliers and leads to a reduced-order system, which is solved using the biconjugate gradient method (BiCGM). It is shown that this method is highly scalable and is more efficient on parallel platforms when solving large matrices than traditional iterative methods such as a preconditioned conjugate gradient algorithm. This is especially true when a perfectly matched layer (PML) absorbing medium is used to terminate the problem domain. <s> BIB004
Algorithms with non overlapping domains involve Schur complement systems. If the system matrix is made of four blocks, the Schur complement matrix S is defined by: It is then shown that solving the global system leads first to solve the Schur system: The unknowns located on the interface separating two neighbouring subdomains give the Schur complement system. Since the subdomains Ω i become then disconnected, the corresponding block matrices A i become also disconnected, allowing an easy parallelization ( fig. 8 ). The Schur matrix system is solved by the CG method for symmetric definite matrix or more generally by Krylov subspace method such as GMRES. Parallel interface preconditioners are often introduced to improve the convergence. Many ways for preconditioning exist in the literature for 2D problems. Preconditioners which not destroy the original scalability and which are not expensive are complicated to obtain for complex geometries and irregular discretizations. Fig. 8 . Basic idea of the Schur complement system method. A smart manner to obtain non overlapping domain decomposition algorithm showing a good parallel efficiency is to enforce the continuity of boundary conditions by introducing Lagrange multipliers BIB003 . The FE Tearing and Interconnecting method is an important class of such hybrid methods BIB001 . It can be viewed as a dual Schur Complement method . The Lagrange multipliers are solution of a small-size linear system compared to the global problem size. When solved with an iterative method (CG or BiCG method), the corresponding matrix does not need to be explicitly assembled. Each local field problem can be treated independently on each processor. In  the resulting algorithm for the solution of the vector wave equation discretized with tetrahedral elements is presented . It is used to modelize 3D PEC wedge in a rectangular cavity. The continuity of tangential electric fields on the interface between subdomains is used to build the Schur system. This system is calculated using parallel conjugate gradient, while the local matrices A i factorized only once. From Fig. 9 , this algorithm is shown to be highly scalable. An other interesting advantage of the FETI method follows from BIB004 . For the solution of Maxwell's equation in the frequency domain, its performance is less affected by the presence of perfectly matched layers solution than a direct iterative method (a divide-and-conquer-type preconditioned BiCGM). On the other hand, the convergence of the method is sensitive to irregular subdomains. The question of constructing efficient parallel preconditionners is also a remaining problem for such a method. Fig.9 . Speedup of the FETI algorithm for the computation of a PEC wedge into a cavity BIB001 . Computation is performed on a 32-node Intel iSPC/860 hypercube. The domain decomposition approach offers many opportunities for the coupling of boundary element method (BEM) and FEM. An unified variational formulation has been proposed in  . In BIB002  , the z-component of the vector potential is determined from the solution of linear, symmetric but indefinite system of coupled equations which are reformulated as symmetric and positive definite systems. A different way to exploit the BEM-FEM coupling is presented in  for 3D transient problem. In this case, the coupling with the BEM is made necessary due to the presence of moving part into the electromechanical device. The developed method is based on a Dirichlet-Neumann algorithm: sequential solutions of the FEM part are linked within a preconditioned GMRES method together with parallel solution of the BEM part ( fig. 10 ). On a NEC-SX4 with 16 processors, the obtained speedup is about 6. 