The Locally Corrected Nyström Method for Electromagnetics

The focus of this chapter is to provide a broad overview of the high-order Locally Corrected Nystrom (LCN) method. The LCN method is applied herein to the solution of a general hybrid surface/volume integral equation defining the electromagnetic scattering and/or radiation from arbitrary three-dimensional geometries of arbitrary material composition. The chapter starts out by providing a basic understanding of the LCN algorithm and its analogy to the more popular method of moments. The LCN method is then specialized to the solution of hybrid integral equations by providing nuts-and-bolts details regarding the application of the method. LCN discretizations of surface integral equations are presented for curvilinear quadrilateral or triangular meshes. Furthermore, LCN discretizations of volume integral equations are derived for curvilinear hexahedral, tetrahedral, or prism meshes. Quadrature rules suitable for a Nystrom discretization and basis functions suitable for local corrections are defined for each topological type. Performing the local corrections for the different kernels encountered is also discussed. Finally a number of numerical examples are presented that span the different integral operators and topological types. For all cases, data is provided that demonstrates the high-order convergence of a LCN solution method. It is hoped that the power, yet simplicity, of the LCN method is conveyed in this chapter, along with enough details for an interested reader to develop their own LCN software.