[pt] Três métodos de eliminação de modos espúrios em soluções de guias de ondas dielétricos, inomogêneos e anisotrópicos, usando formulações variacionais e o Método de Elementos Finitos são analisados. O método das penalidades com a técnica de integração reduzida seletiva é aplicado pela primeira em problemas de Eletromagnetismo. Através da análise de vários exemplos, seus resultados são comparados aos obtidos com o método das componentes transversais devido a Hayata e Koshiba e com o uso de elementos de aresta. A qualidade das aproximações e o desempenho computacional comprovam a eficiência da integração reduzida, que eliminou os principais incovenientes do método das penalidades, mantendo seus atrativos e a simplicidade da implementação. O uso de elementos de aresta também se mostrou uma abordagem atraente embora seus mecanismos de funcionamento ainda não estejam completamente entendidos e apesar da maior complexidade na implementação. / [en] Three methods for elimination of spurious modes from variationally formulated Finite Element solutions of inhomogeneous/anisotropic dieletric waveguides are compared. The Reduced Integration Penalty Method is applied for the first time. To EM wave problems. From the analyses of various examples, the results of this method are compared to those obtained with the Hayeta and Koshiba transversal component method and with the use of edge elements. The quality of the approximations and the computacional performance testify the efficacy of the reduced integration that eliminates the main drawbacks fo the Penalty method while preserving its advantages and simplicity of implementation. The use of edge elements was proven to be a very attractive approach, although its mechanisms are still to be fully understood and its implementation may not always be straightforward in a standard Finite

Maxwell's equations are a system of partial differential equations of vector fields. Imposing the constitutive relations for material properties yields equations for the curl and divergence of the electric and magnetic fields. The curl and divergence equations must be solved simultaneously, which is not the same as solving three separate scalar problems in each component of the vector field. This thesis describes a new method for solving partial differential equations of vector fields using the finite element method. New basis functions are used to solve the curl equation while allowing the divergence to be set as a constraint. The basis functions are defined on a mesh of bricks and the method is applicable for geometries that conform to a Cartesian coordinate system. The basis functions are a combination of cubic Hermite splines and second order Lagrange interpolation polynomials. The method yields a linearly independent set of constraints for the divergence, which is modelled to second order accuracy within each brick. Mesh refinement is accomplished by dividing selected bricks into $2\times 2\times 2$ smaller bricks of equal size. The change in the node pattern at an interface where mesh refinement occurs necessitates a modified implementation of the divergence constraints as well as additional constraints for hanging nodes. The mesh can be refined to an arbitrary number of levels. The basis functions can exactly model the discontinuity in the normal component of the field at a planar interface. The method is modified to solve problems with singularities at material boundaries that form $90^{\circ}$ edges and corners. The primary test problem of the new basis functions is to obtain the resonant frequencies and fields of three-dimensional cavities. The new basis functions can resolve physical solutions and non-physical, spurious modes. The eigenvalues obtained with the new method are in good agreement with exact solutions and experimental values in cases where they exist. There is also good agreement with results from second-order edge elements that are obtained with the software package HFSS. Finally, the method is modified to solve problems in cylindrical coordinates provided the domain does not contain the coordinate axis.

Maxwell's equations are a system of partial differential equations of vector fields. Imposing the constitutive relations for material properties yields equations for the curl and divergence of the electric and magnetic fields. The curl and divergence equations must be solved simultaneously, which is not the same as solving three separate scalar problems in each component of the vector field. This thesis describes a new method for solving partial differential equations of vector fields using the finite element method. New basis functions are used to solve the curl equation while allowing the divergence to be set as a constraint. The basis functions are defined on a mesh of bricks and the method is applicable for geometries that conform to a Cartesian coordinate system. The basis functions are a combination of cubic Hermite splines and second order Lagrange interpolation polynomials. The method yields a linearly independent set of constraints for the divergence, which is modelled to second order accuracy within each brick. Mesh refinement is accomplished by dividing selected bricks into $2\times 2\times 2$ smaller bricks of equal size. The change in the node pattern at an interface where mesh refinement occurs necessitates a modified implementation of the divergence constraints as well as additional constraints for hanging nodes. The mesh can be refined to an arbitrary number of levels. The basis functions can exactly model the discontinuity in the normal component of the field at a planar interface. The method is modified to solve problems with singularities at material boundaries that form $90^{\circ}$ edges and corners. The primary test problem of the new basis functions is to obtain the resonant frequencies and fields of three-dimensional cavities. The new basis functions can resolve physical solutions and non-physical, spurious modes. The eigenvalues obtained with the new method are in good agreement with exact solutions and experimental values in cases where they exist. There is also good agreement with results from second-order edge elements that are obtained with the software package HFSS. Finally, the method is modified to solve problems in cylindrical coordinates provided the domain does not contain the coordinate axis.