An Iteration-Free Domain Decomposition Method for the Fast Finite Element Analysis of Electromagnetic Problems

Abstract

An iteration-free finite element domain decomposition method (DDM) is proposed for the simulation of 3-D electromagnetic problems. The finite element tearing and interconnecting (FETI) with Robin-type transmission condition is introduced to construct a domain decomposition framework. Based on hierarchical ( $\mathcal {H}$ -) matrix algorithm, the numerical Green’s function of each subdomain can be represented by a data-sparse matrix form. Subsequently, the original 3-D problem is reduced to a 2-D problem on the skeleton. Derived from the finite element discretization of pseudo-differential operator, the skeleton matrix can also be efficiently approximated with an $\mathcal {H}$ -matrix by assembling all the numerical Green’s functions of subdomains. Based on $\mathcal {H}$ -LU factorization algorithm, the computational complexity and memory requirement for the direct solution of the skeleton equation can be reduced to be logarithmic linear. The resulting direct FETI (D-FETI) method completely depends on direct method without any iterative process. Hence, it does not suffer from the issue of convergence rate, and avoids redundant computations for multiple right-hand side problems. Numerical examples demonstrate the effectiveness of the proposed method.