An integral equation technique for the exterior and interior Neumann problem in toroidal regions (Q1077146)

The paper presents an integral equation method for general three- dimensional toroidal geometry D with Neumann boundary conditions prescribed on the boundary \(\partial D\) of D. By means of Green's theorem the Laplace equation for \(\phi\) is converted into an integral equation \[ (1)\quad \phi (x)+1/2\pi \int_{\partial D}df'\partial G(x,x')/\partial n'\phi (x')=1/2\pi \int_{\partial D}df'G\quad (x,x')\partial \phi (x')/\partial n' \] which is recast in terms of the angle-like independent variables u and v to give the form \[ (2)\quad \phi (u,v)+\int^{1}_{0}\int^{1}_{0}du'dv'g(u,v,u',\quad v')\phi (u',\quad v')=\int^{1}_{0}\int^{1}_{0}du'dv'\quad h(u,v,u',\quad v')... \] The potential \(\phi\) (u,v) is a periodic function of u and v hence the authors express \(\phi\) as a Fourier series \[ (3)\quad \phi (u,v)=\sum^{\infty,\infty}_{m=-\infty,n=-\infty}{\hat \phi}_{mn}e^{2\pi i(mu+nv)}, \] With (3) substituted in (2), the Fourier transform with respect to u and v leads to an infinite set of linear equations of the Fourier coefficients \({\hat \phi}{}_{mn}\). The singularity involved in the kernel g and the source term h of (2) is treated by a regularization method for numerical approximation. The paper gives a very impressive set of numerical results based on the author's computer code named NESTOR.