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In this paper, the biorthonormal-basis method has been used to model the complex wave-propagation constant and the transversefield pattern in inhomogeneously filled waveguides with lossless and lossy dielectrics. The differential operator governing the transverse fields is transformed into a linear-matrix eigenvalue problem, using the eigenvectors of an auxiliary problem to expand the modes of the original problem. This method has been applied to the calculation of the complex modes in dielectric-rod-loaded circular waveguides and dielectric-slab-loaded rectangular waveguides, which provides an example demonstrating the capability of the method to include the dielectric losses directly in its formulation. The method is free of spurious modes and, in most cases (as in the systems shown in this paper), the integrals involved in the matrix elements can be obtained analytically, and the only numerical approximation is the finite number of modes used for the expansion of the fields. The high accuracy necessary to obtain complex modes is fully achieved by our method. On the contrary, other methods involve serious difficulties. ACKNOWLEDGEMENTThis work was financially supported by the Ministerio de Ciencia y Tecnología (grant TIC2000-0591-C03-03), Spain. INTRODUCTIONInclined slots in the narrow wall of a rectangular waveguide, so-called edge slots, have been widely used in radars and satellite systems. These slots extend onto the broad walls of the waveguide to produce resonance, which complicates the analysis of narrow wall slots. Furthermore, fabrication of planar arrays become difficult, since metallic spacers should be placed in between the guides, which in turn increases the back radiation of the array. A number of investigations have been reported on these slots [1][2][3][4]. In all these works, the edge slot is tilted to excite and penetrate the broad walls of the waveguide. Hashemi-Yeganeh and Elliott [5] analyzed untilted edge slots excited by tilted wires in a rectangular waveguide. One of the most important results they obtained in their study was the discovery of the possibility of a resonant condition between the slot and tilted wires, which does not require the slot to be continued onto the broad walls. This permits embedding an array of these composite elements in a ground plane. In their analysis it was assumed that the slot is embedded in an infinite ground plane. Hirokawa et al. [6] analyzed this structure by including the actual structure, using a spectrum of the two-dimensional solutions (S2DS) method [7]. Their results show that the effect of modelling the actual outer cross section of the waveguide, instead of assuming an infinite ground plane, is small. Actually, the difference is smaller than the measurement error in the experimental setup. Hirokawa and Kildal another excitation technique by inserting a dielectric plate into the slot on which conducting strips are etched. Again, a resonance can occur which does not require the slot to be continued onto the broad walls. These two methods of exc...

This paper presents a systematic approach to constructing high-order tangential vector basis functions for the multilevel finite element solution of electromagnetic wave problems. The new bases allow easy computation of a preconditioner to eliminate or at least weaken the indefiniteness of the system matrix and thus reduce the condition number of the system matrix. When these bases are used in multilevel solutions, where the multilevels correspond to the order of the basis functions, the resulting p-multilevel-ILU preconditioned conjugate gradient method (MPCG) provides an optimal rate of convergence. We first derive an admissible set of vectors of order p, and decompose this set into two subspaces-rotational and irrotational (gradient). We then reduce the number of vectors by making them orthogonal to all previously constructed lower-order bases. The remaining vectors are made mutually orthogonal in both the vector space and in the range space of the curl operator. The resulting vector basis functions provide maximum orthogonality while maintaining tangential continuity of the field. The zeroth-order space is further decomposed using a scalar-vector formulation to eliminate convergence problems at extremely low frequencies.Numerical experiments show that number of iterations needed for the solution by MPCG is basically constant, regardless of the order of the basis or of the matrix size. Computational speed is improved by several orders of magnitude due to the fast matrix solution of MPCG and to the high accuracy of the higher-order bases.Key words. P -multilevel finite element methods, Maxwell's equations, multilevel preconditioned conjugate gradient method, tangential vector basis functions AMS subject classifications. 65N22, 65N55, 65F10, 83C50 PII. S1064827500367531 1. Introduction. In 1980, Nedelec defined a set of vector finite element basis functions having the properties that they are complete to order p in the range space of the curl operator, that they impose tangential but not normal continuity of the vector, and that they are unisolvent [1].In this paper, we will call finite elements based on these functions Nedelec elements and denote the function space spanned by these elements as E p (curl). Nedelec elements have been shown to be important in the solution of electromagnetic field problems [2], [3]. In particular, [4], [5], [6], [7] showed that these elements eliminate the problem of spurious modes that plague conventional node based approximations of the vector wave equation derived from Maxwell's equations. Numerous authors have derived alternative forms of the Nedelec bases, both for low-order "edge elements" [3], [7], [8], [9], [10], [11], [12] and for high-order tangential vector elements [5], [6], [13], [14], [15], [16], [17], [18], [19], [20], [21]. The goal of these constructions has often been to make the Nedelec bases interpolatory or hierarchical. Hiptmair [22] recently provided a general abstract framework for the systematic construction of these vector bases. With such an abundance of ...

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Finite-element vector potential solutions of three-dimensional magnetic field problems are usually obtained by approximating each component of the vector potential by a separate set of scalar finite-element approximation functions and by imposing continuity conditions between elements on all three components. This procedure is equivalent to imposing continuity of both the normal and the tangential components of the vector potential. We show in this paper that this procedure is too restrictive: While continuity of the tangential component of the vector potential is required, continuity of the normal components is not essential in the variational formulation. We introduce a new type of vector finite-element approximation function that has the property that it interpolates not to point values of each component of vector potential, but rather to the tangential projection of the vector potential on each edge of tetrahedral finite elements. With the new basis functions, continuity of the normal component of the vector potential is provided only approximately by means of the natural interface conditions inherent in the variational procedure. This results in a more efficient procedure for the solution of three-dimensional magnetostatic field problems than is obtained by enforcing normal component continuity exactly.

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