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Abstract-The Poincaré-Steklov operator provides a direct relation between the tangential electric and magnetic field at the boundary of a simply connected domain, and a discrete equivalent of the operator can be constructed from the sparse finite element (FE) matrix of that domain by forming the Schur complement to eliminate the interior unknowns. Identifying the FE system matrix as a discretized version of the Poincaré-Steklov operator allows us to describe and analyze FE and hybrid finite element-boundary integral equation (FE-BIE) formulations from an operator point of view. We show how this operator notation provides substantial theoretical insight into the analysis of spurious solutions in hybrid FE-BIE methods, and we apply the theory on a TM scattering example to predict the breakdown frequencies of different hybrid formulations. Index Terms-Hybrid methods, Electromagnetic scattering I. INTRODUCTIONThe hybrid FE-BIE method is a widely used approach to numerically solve electromagnetic scattering or radiation problems. It combines the versatility of the FE method to model complex inhomogeneous and anisotropic structures with the accuracy and efficiency of the BIE method to model large homogeneous and potentially unbounded domains. Traditionally, the FE formulation is set in a variational framework, which is extended with the BIE formalism to form a hybrid variational formulation. However, this approach relies on the internal field densities in the FE domain, and since the exact method used to couple both formulations seems to be very important to avoid spurious solutions, we expect that a hybrid formalism specifically focusing on the behavior of the formulations at the boundary will provide more theoretical insights.In this paper, we use the concept of a Poincaré-Steklov (PS) operator to describe the FE formulation in a domain by the relation it provides between the tangential electric and magnetic fields at the boundary of the domain. Hybrid formulations are easily formed by properly combining the PS and BIE operators. Unlike the classical variational framework, this operator notation retains the individual identities of the FE and BIE operators in FE-BIE formulations, and different properties regarding spurious solutions are easily derived. After outlining the general equations in Section II, we use the new operator notation in Section III to analyze the problem of spurious solutions in commonly used hybrid FE-BIE formulations. The theoretical results are then applied in Section IV to compare the analytical and the simulated breakdown frequencies of the different FE-BIE formulations for a TM scattering problem involving a single dielectric cylinder.

Abstract-This article presents a hybrid finite elementboundary integral equation (FE-BIE) method where the boundary integral interactions containing the 2D Green's kernel function are accelerated by the nondirective stable plane wave multilevel fast multipole algorithm (NSPW-MLFMA). This hybrid method enables the fast simulation of very large scale scattering problems with multiple homogeneous and inhomogeneous dielectrics and perfectly electric conducting (PEC) objects. The new hybrid technique with FMM acceleration applies for both high frequency as low frequency as long as the finite element mesh is sufficiently fine to contain the numerical dispersion within the desired accuracy. The hybrid formulation is outlined, and its validity is demonstrated by means of a 2D scattering problem.

Abstract-The iterative solution of the hybrid finite elementboundary integral equation (FE-BIE) system can be accelerated by using a multilevel fast multipole algorithm (MLFMA) to accelerate the BIE interactions. The far interactions in the MLFMA are of low complexity, which can be traced back to the original dense BIE matrix where off-diagonal submatrices are of low-rank, provided the unknowns are properly ordered. To accelerate the FE part, we investigate the existence of similar low-rank properties in the FE matrix. Previously, they were shown to exist in the FE discretization matrices of elliptic partial differential equations, but not for the non-elliptic time harmonic system of Maxwell's equations. Comparing the Schur complement form of the FE system with a discrete Poincaré-Steklov operator, the existence of such low-rank properties is suggested, and this theory is confirmed experimentally by means of a numerical example.

Through careful selection of the penalty factors lambda n and keeping the constraint function piecewise linear close to the boundary of the feasible region (p = 1), we were able to transform the constrained problem into an unconstrained one. Compared to the Lagrange multiplier method, far more complex problems can be solved. Simulations confirm that this new approach yields valid results