New trends in geometric modeling and discretization for integral equations

In this chapter, the author have presented ideas along these lines, focusing on two different numerical approaches, i.e., GMM and IGA, both of which rely on subdivision representation of geometries. Both methods take different approaches to solving integral equations. GMM is a highly flexible scheme that permits the use of different basis functions for each patch and, as a result, is highly customizable. The crux to this approach is local surface parameterization and transition maps between different local parameterizations in regions where patches overlap. Subdivision offers an effective approach to overcome this bottleneck. Its efficacy and related challenges have been demonstrated through examples. Indeed, it is possible to pair subdivision GMM with methods developed in [14] to efficiently evaluate integrals to solve problems that are electrically large and geometrically complex.