An algebraic domain decomposition algorithm for the vector finite-element analysis of 3D electromagnetic field problems

This Letter, proposes an algebraic domain decomposition algorithm (ADDA) to solve large sparse linear systems derived from the vector finite-element method (FEM) for 3D electromagnetic field problems. The proposed method segments the problem into several smaller pieces, solves each subproblem by direct methods, and then reassembles the subproblem solutions together to get the global result. Block LU factorization and multifrontal method are applied to solve each subproblem for the generation of the reduced system, and iterative methods are applied to solve the reduced system. It is shown that if combined with ADDA, biconjugate gradient method (BCG) converges more rapidly than the conjugate gradient method (CG), and both of them are faster than the conventional CG method. The simulation results demonstrate that the proposed algorithm can efficiently solve large and sparse linear equations arising from the finite-element method for the electromagnetic problems involving complex media such as perfectly matched layers (PMLs), which often make the linear equation ill-conditioned.