  The state of art of electromagnetic integral equations has seen significant
growth over the past few decades, overcoming some of the fundamental
bottlenecks: computational complexity, low frequency and dense discretization
breakdown, preconditioning, and so on. Likewise, the community has seen
extensive investment in development of methods for higher order analysis, in
both geometry and physics. Unfortunately, these standard geometric descriptors
are continuous, but their normals are discontinuous at the boundary between
triangular tessellations of control nodes, or patches, with a few exceptions;
as a result, one needs to define additional mathematical infrastructure to
define physical basis sets for vector problems. In stark contrast, the
geometric representation used for design are second order differentiable almost
everywhere on the surfaces. Using these description for analysis opens the door
to several possibilities, and is the area we explore in this paper. Our focus
is on Loop subdivision based isogeometric methods. In this paper, our goals are
two fold: (i) development of computational infrastructure for isogeometric
analysis of electrically large simply connected objects, and (ii) to introduce
the notion of manifold harmonics transforms and its utility in computational
electromagnetics. Several results highlighting the efficacy of these two
methods are presented.
