Thesis (MEng)--Stellenbosch University, 2015. / ENGLISH ABSTRACT: The Method of Moments (MoM) is routinely used for the numerical solution of electromagnetic surface integral equations. Solution errors are inherent to any numerical computational method, and error estimators can be effectively employed to reduce and control these errors. In this thesis, gradient recovery techniques of the Finite Element Method (FEM) are formulated within the MoM context, in order to recover a higher-order charge of a Rao-Wilton-Glisson (RWG) MoM solution. Furthermore, a new recovery procedure, based specifically on the properties of the RWG basis functions, is introduced by the author. These recovered charge distributions are used for a posteriori error estimation of the charge. It was found that the newly proposed charge recovery method has the highest accuracy of the considered recovery methods, and is the most suited for applications within recovery based error estimation. In addition to charge recovery, the possibility of recovery procedures for the MoM solution current are also investigated. A technique is explored whereby a recovered charge is used to find a higher-order divergent current representation. Two newly developed methods for the subsequent recovery of the solenoidal current component, as contained in the RWG solution current, are also introduced by the author. A posteriori error estimation of the MoM current is accomplished through the use of the recovered current distributions. A mixed second-order recovered current, based on a vector recovery procedure, was found to produce the most accurate results. The error estimation techniques developed in this thesis could be incorporated into an adaptive solver scheme to optimise the solution accuracy relative to the computational cost. / AFRIKAANSE OPSOMMING: Die Moment Metode (MoM) vind algemene toepassing in die numeriese oplossing van elektromagnetiese oppervlak integraalvergelykings. Numeriese foute is inherent tot die prosedure: foutberamingstegnieke is dus nodig om die betrokke foute te analiseer en te reduseer. Gradiënt verhalingstegnieke van die Eindige Element Metode word in hierdie tesis in die MoM konteks geformuleer. Hierdie tegnieke word ingespan om die oppervlaklading van 'n Rao-Wilton-Glisson (RWG) MoM oplossing na 'n verbeterde hoër-orde voorstelling te neem. Verder is 'n nuwe lading verhalingstegniek deur die outeur voorgestel wat spesifiek op die eienskappe van die RWG basis funksies gebaseer is. Die verhaalde ladingsverspreidings is geïmplementeer in a posteriori fout beraming van die lading. Die nuut voorgestelde tegniek het die akkuraatste resultate gelewer, uit die groep verhalingstegnieke wat ondersoek is. Addisioneel tot ladingsverhaling, is die moontlikheid van MoM-stroom verhalingstegnieke ook ondersoek. 'n Metode vir die verhaling van 'n hoër-orde divergente stroom komponent, gebaseer op die verhaalde lading, is geïmplementeer. Verder is twee nuwe metodes vir die verhaling van die solenodiale komponent van die RWG stroom deur die outeur voorgestel. A posteriori foutberaming van die MoM-stroom is met behulp van die verhaalde stroom verspreidings gerealiseer, en daar is gevind dat 'n gemengde tweede-orde verhaalde stroom, gebaseer op 'n vektor metode, die beste resultate lewer. Die foutberamingstegnieke wat in hierdie tesis ondersoek is, kan in 'n aanpasbare skema opgeneem word om die akkuraatheid van 'n numeriese oplossing, relatief tot die berekeningskoste, te optimeer.

O trabalho aborda a questão das estimativas a posteriori dos erros de discretização e particularmente a recuperação dos gradientes de soluções numéricas obtidas com o método dos elementos finitos (MEF) e com o método dos elementos finitos generalizados (MEFG). Inicialmente, apresenta-se, em relação ao MEF, um resumido estado da arte e conceitos fundamentais sobre este tema. Em seguida, descrevem-se os estimadores propostos para o MEF denominados Estimador Z e \"Superconvergent Patch Recovery\" (SPR). No âmbito do MEF propõe-se de modo original a incorporação do \"Singular Value Decomposition\" (SVD) ao SPR aqui mencionada como SPR Modificado. Já no contexto do MEFG, apresenta-se um novo estimador do erro intitulado EPMEFG, estendendo-se para aquele método as idéias do SPR Modificado. No EPMEFG, a função polinomial local que permite recuperar os valores nodais dos gradientes da solução tem por suporte nuvens (conjunto de elementos finitos que dividem um nó comum) e resulta da aplicação de um critério de aproximação por mínimos quadrados em relação aos pontos de superconvergência. O número destes pontos é definido a partir de uma análise em cada elemento que compõe a nuvem, considerando-se o grau da aproximação local do campo de deslocamentos enriquecidos. Exemplos numéricos elaborados com elementos lineares triangulares e quadrilaterais são resolvidos com o Estimador Z, o SPR Modificado e o EPMEFG para avaliar a eficiência de cada estimador. Essa avaliação é realizada mediante o cálculo dos índices de efetividade. / The paper addresses the issue of a posteriori estimates of discretization errors and particularly the recovery of gradients of numerical solutions obtained with the finite element method (FEM) and the generalized finite element method (GFEM). Initially, it is presented, for the MEF, a brief state of the art and fundamental concepts about this topic. Next, it is described the proposed estimators for the FEM called Z-Estimator and Superconvergent Patch Recovery (SPR). It is proposed, originally, in the ambit of the FEM, the incorporation of the \"Singular Value Decomposition (SVD) to SPR mentioned here as Modified SPR. On the other hand, in the context of GFEM, it is presented a new error estimator entitled EPMEFG in order to expand the ideas of Modified SPR to that method. In EPMEFG, the local polynomial function that allows to recover the nodal values of the gradients of the solution has for support clouds (set of finite elements that share a common node) and results from the applying of a criterion of least squares approximation in relation to the superconvergent points. The number of these points is defined from an analysis of each cloud\'s element, considering the degree of local approximation of the displacement field enriched. Numerical examples elaborated with linear triangular and quadrilateral elements are solved with the Z-Estimator, the Modified SPR and the EPMEFG to evaluate the efficiency of each estimator. This evaluation is done calculating the effectivity indexes.

O trabalho aborda a questão das estimativas a posteriori dos erros de discretização e particularmente a recuperação dos gradientes de soluções numéricas obtidas com o método dos elementos finitos (MEF) e com o método dos elementos finitos generalizados (MEFG). Inicialmente, apresenta-se, em relação ao MEF, um resumido estado da arte e conceitos fundamentais sobre este tema. Em seguida, descrevem-se os estimadores propostos para o MEF denominados Estimador Z e \"Superconvergent Patch Recovery\" (SPR). No âmbito do MEF propõe-se de modo original a incorporação do \"Singular Value Decomposition\" (SVD) ao SPR aqui mencionada como SPR Modificado. Já no contexto do MEFG, apresenta-se um novo estimador do erro intitulado EPMEFG, estendendo-se para aquele método as idéias do SPR Modificado. No EPMEFG, a função polinomial local que permite recuperar os valores nodais dos gradientes da solução tem por suporte nuvens (conjunto de elementos finitos que dividem um nó comum) e resulta da aplicação de um critério de aproximação por mínimos quadrados em relação aos pontos de superconvergência. O número destes pontos é definido a partir de uma análise em cada elemento que compõe a nuvem, considerando-se o grau da aproximação local do campo de deslocamentos enriquecidos. Exemplos numéricos elaborados com elementos lineares triangulares e quadrilaterais são resolvidos com o Estimador Z, o SPR Modificado e o EPMEFG para avaliar a eficiência de cada estimador. Essa avaliação é realizada mediante o cálculo dos índices de efetividade. / The paper addresses the issue of a posteriori estimates of discretization errors and particularly the recovery of gradients of numerical solutions obtained with the finite element method (FEM) and the generalized finite element method (GFEM). Initially, it is presented, for the MEF, a brief state of the art and fundamental concepts about this topic. Next, it is described the proposed estimators for the FEM called Z-Estimator and Superconvergent Patch Recovery (SPR). It is proposed, originally, in the ambit of the FEM, the incorporation of the \"Singular Value Decomposition (SVD) to SPR mentioned here as Modified SPR. On the other hand, in the context of GFEM, it is presented a new error estimator entitled EPMEFG in order to expand the ideas of Modified SPR to that method. In EPMEFG, the local polynomial function that allows to recover the nodal values of the gradients of the solution has for support clouds (set of finite elements that share a common node) and results from the applying of a criterion of least squares approximation in relation to the superconvergent points. The number of these points is defined from an analysis of each cloud\'s element, considering the degree of local approximation of the displacement field enriched. Numerical examples elaborated with linear triangular and quadrilateral elements are solved with the Z-Estimator, the Modified SPR and the EPMEFG to evaluate the efficiency of each estimator. This evaluation is done calculating the effectivity indexes.