A vector dual-primal finite element tearing and interconnecting method for solving 3-D large-scale electromagnetic problems

A Lagrange multiplier based non-overlapping domain decomposition method, referred to. as the dual-primal finite element tearing and interconnecting (FETI-DP), is formulated for the finite element simulation of large, three-dimensional (3-D) electromagnetic problems. This formulation extends the FETI-DP for solving the scalar Helmholtz equation to the solution of the vector curl-curl wave equation using edge-based finite elements. It enforces the field continuity explicitly along the edges shared by more than two subdomains and implicitly at the interfaces between two subdomains through the use of Lagrange multipliers. With the aid of a direct sparse solver for each subdomain system, the large global problem is reduced to a much smaller interface problem, from which a Neumann boundary condition is obtained at the interfaces between all the subdomains. This Neumann boundary condition is then used to calculate the field within each subdomain. It is shown that the resulting FETI-DPEM method is scalable with respect to the size of finite elements and the number of subdomains. It is also scalable with respect to the size of the subdomains when the subdomains, with its surfaces enclosed by perfect magnetic conductors, cannot support any resonant modes. The FETI-DPEM method is applied to the electromagnetic simulation of array-type structures where the geometrical redundancy is fully exploited to speedup the simulation and reduce the memory requirement. Numerical results for the simulation of finite antenna arrays and photonic bandgap devices are presented to demonstrate the application, accuracy, efficiency, and capability of the FETI-DPEM method.