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In this dissertation a generalization of the locally corrected Nyström (LCN) discretization method is outlined wherein sparse transformations of the LCN system matrix are obtained via singular value decompositions of local constraint matrices. The local constraint matrices are used to impose normal continuity of the currents across boundaries shared by mesh elements. For this reason, the method is called constrained LCN (CLCN). Due to the CLCN’s simplicity and flexibility, it is straightforward to develop high order CLCN systems for different formulations and mesh element types. As compared to the LCN, the CLCN method offers memory savings and improved accuracy when applied to geometries with sharp edges. Furthermore, the CLCN method maintains the high-order convergence of the LCN method, and it eliminates the need to include line charges in Nyström-based discretizations of formulations that involve the continuity equation. In addition to developing the CLCN method, we will investigate a high-order Nyström implementation of an augmented VIE called AVIE as an alternative to the VIE formulation. The limitations of VIE include poor matrix condition numbers for problems with high contrast materials, and deteriorating performance for problems with complex, multi-scale meshes. The AVIE formulation incorporates surface and volume charges as additional unknowns and includes both current continuity and charge neutrality constraints. It is found that a straightforward LCN implementation of the AVIE formulation is still poorly conditioned in some cases. However, it will be shown that the AVIE formulation provides sufficient flexibility to enable the resulting linear system to be appropriately scaled in order to significantly improve the matrix condition numbers.