In this paper we present a numerically stable method for the model order reduction of finite element (FE) approximations to passive microwave structures parameterized by polynomials in several variables. The proposed method is a projection-based approach using Krylov subspaces and extends the works of Gunupudi et al. (P. Gunupudi, R. Khazaka and M. Nakhla, Analysis of transmission line circuits using multidimensional model reduction techniques, IEEE Trans. Adv. Packaging 25 (2002), pp. 174-180) and Slone et al. (R.D. Slone, R. Lee and J.-F. Lee, Broadband model order reduction of polynomial matrix equations using single-point well-conditioned asymptotic waveform evaluation: derivations and theory, Int. J. Numer. Meth. Eng. 58 (2003), pp. 2325-2342). First, we present the multivariate Krylov space of higher order associated with a parameter-dependent righthand-side vector and derive a general recursion for generating its basis. Next, we propose an advanced algorithm to compute such a basis in a numerically stable way. Finally, we apply the Krylov basis to construct a reduced order model of the moment-matching type. The resulting single-point method requires one matrix factorization only. Numerical examples demonstrate the efficiency and reliability of our approach.

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An enhanced finite-element method (FEM) for the vector wave equation is presented. For improved speed and stability ranging from microwave frequencies down to the static limit, we propose a multilevel solver that uses a tree-gauged formulation on the coarsest mesh and a partially gauged scheme for the iterative cycle. Moreover, we have generalized the concept of hanging nodes to higher order (curl)-conforming tetrahedral elements. The combination of hierarchical basis functions and the hanging variables framework yields great flexibility in placing degrees of freedom and provides a very attractive alternative to remeshing in an -adaptive context.Index Terms-Electromagnetic (EM) fields, finite-element methods (FEMs), wave propagation.

We propose a systematic methodology for the construction of hanging variables to connect finite elements of unequal refinement levels within a nested tetrahedral mesh. While conventional refinement schemes introduce irregular elements at such interfaces which must be removed when the mesh is further refined, the suggested approach keeps the discretization perfectly nested. Thanks to enhanced regularity, mesh-based methods such as refinement algorithms or intergrid transfer operators for use in multigrid solvers can be implemented in a much simpler fashion. This paper covers 1 and (curl) basis functions for triangular or tetrahedral elements.Index Terms-Edge and facet elements, electromagnetic fields, finite element methods.

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