Introduction
This package provides the Sauter-Schwab regularizing coordinate transformations [1] such that 4D integrals of the form
\[\int_{\Gamma}\int_{\Gamma'}b_i(\bm{x})\,k(\bm{x},\bm{y})\, b_j(\bm{y})\,\mathrm{d}S(\bm{y})\,\mathrm{d}S(\bm{x})\]
with Cauchy-singular integral kernels $k(\bm{x},\bm{y})$ can be integrated via numerical quadrature. The integrals denote double surface integrals over
- triangles (curved or flat) or
- quadrilaterals (curved or flat)
$\Gamma$ and $\Gamma'$ in 3D Space. The functions $b_i(\bm{x})$ and $b_i(\bm{y})$ are assumed to be real valued and non-singular.
These kind of integrals occur in the area of boundary element methods (BEM) for solving elliptic partial differential equations. It can be interpreted as the interaction of the two basisfunctions $b_i(\bm{x})$ and $b_i(\bm{y})$, with respect to their domains $\Gamma$ and $\Gamma'$, which, for instance, correspond to the cells of a meshed surface.
The triangles or quadrilaterals must be either equal, have two vertices in common, have one vertex in common or do not touch at all. A partial overlap is forbidden.
In the current implementation $\Gamma$ and $\Gamma'$ have to be both either triangles or quadrilatersls. However, mixed cases can be implemented, too.
References
[1] Sauter S. Schwab C., "Boundary Element Methods (Springer Series in Computational Mathematics)", Chapter 5, Springer, 2010.