Integral equation formulations to solve electromagnetic scattering and radiation problems have existed for over a century. The method of moments (MoM) technique to solve these integral equations has been in active use for over 40 years and has become one of the cornerstones of electromagnetic analysis. It has been successfully employed in a wide variety of problems ranging from scattering and antenna analysis to electromagnetic compatibility analysis to photonics. In MoM, the unknown quantity (currents or fields) is represented using a set of basis functions. This representation, together with Galerkin testing, results in a set of equations that may then be solved to obtain the coefficients of expansion. The basis functions are typically constructed on a tessellation of the geometry and its choice is critical to the accuracy of the final solution. As a result, considerable energy has been expended in the design and construction of optimal basis functions. The most common of these functions in use today are the Rao-Wilton-Glisson (RWG) functions that have become the de-facto standard and have also spawned a set of higher order complete and singular variants. However, their near-ubiquitous popularity and success notwithstanding, they come with certain important limitations.The chief among these is the intimate marriage between the underlying triangulation of the geometry and the basis function. While this coupling maintains continuity of the normal component of these functions across triangle boundaries and makes them very easy to implement, this also implies an inherent restriction on the kind of basis function spaces that can be employed. This thesis aims to address this issue and provides a novel framework for the discretization of integral equations that demonstrates several significant advantages.In this work, we will describe a new umbrella framework for the discretization of integral equations called the Generalized Method of Moments (GMM). We will show that within this framework it is possible to choose from a very wide variety of functions to form the basis space. While the choice of basis functions can be completely arbitrary, the thesis will describe in detail one particular scheme that utilizes a surface quasi-Helmholtz decomposition and provides a mechanism for the use of any available knowledge of the physics of the problem. The thesis will describe the theoretical framework required for the design of this scheme and develop error bounds for the approximation of the unknown quantities using the proposed basis functions. Following that, we will delve into the implementation of the GMM starting from commonly available triangular tessellations, carefully identifying and solving several key issues along the way.The GMM, developed here, will be applied to a plethora of examples, validated against analytical solutions and compared against the standard RWG discretizations. It will further be shown that the use of the quasi-Helmholtz decomposition will result in a scheme that is well conditioned over a wide range of frequencies. We will then extend the GMM to both the analysis of scattering from dielectric bodies using the PMCHWT and \Muller operators and to the analysis of transient scattering from conducting objects. Several results will be presented in either case that demonstrate the advantage of the proposed method.
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