On the variational formulation of hybrid finite element-boundary integral techniques for electromagnetic analysis

This paper addresses the time-harmonic, electromagnetic analysis of a three-dimensional inhomogeneous radiator/scatterer in free-space. Such analysis can be carried out by combining the finite element method (FEM) with the method of moments (MoM), which yields finite element-boundary integral (FE-BI) formulations. A general framework is presented, within which stationary FE-BI formulations can be established (variational boundary-value problems), which relate to equivalent, underlying variational principles (stationary functionals). The formulations are shown to be accurate, robust and computationally efficient. They avoid the problem of interior resonances without resorting to the combined field integral equation and they result in symmetric system matrices, which preserve reciprocity explicitly. Thus, the stationary FE-BI framework combines the FEM and MoM on the continuous level, as opposed to the usual approach of hybridization after discretization, which generally leads to asymmetric matrices. The stationary FE-BI framework allows one to solve either for the electric and magnetic fields on the volume of the problem domain, or for one volume and one exterior surface field quantity, with only marginal differences in computational cost. The volume-surface formulations have the same storage requirements as previous FE-BI formulations and can be more efficiently solved. The volume-volume formulations provide simultaneous solutions of the electric and magnetic fields, which could for instance be used to construct error estimators directly based on Maxwell's equations.