Spectral method for matching exterior and interior elliptic problems

Abstract

A spectral method is described for solving coupled elliptic problems on an interior and an exterior domain. The method is formulated and tested on the two-dimensional interior Poisson and exterior Laplace problems, whose solutions and their normal derivatives are required to be continuous across the interface. A complete basis of homogeneous solutions for the interior and exterior regions, corresponding to all possible Dirichlet boundary values at the interface, are calculated in a preprocessing step. This basis is used to construct the influence matrix which serves to transform the coupled boundary conditions into conditions on the interior problem. Chebyshev approximations are used to represent both the interior solutions and the boundary values. A standard Chebyshev spectral method is used to calculate the interior solutions. The exterior harmonic solutions are calculated as the convolution of the free-space Green's function with a surface density; this surface density is itself the solution to an integral equation which has an analytic solution when the boundary values are given as a Chebyshev expansion. Properties of Chebyshev approximations insure that the basis of exterior harmonic functions represents the external near-boundary solutions uniformly. The method is tested by calculating the electrostatic potential resulting from charge distributions in a rectangle. The resulting influence matrix is well-conditioned and solutions converge exponentially as the resolution is increased. The generalization of this approach to three-dimensional problems is discussed, in particular the magnetohydrodynamic equations in a finite cylindrical domain surrounded by a vacuum

Poloidal–toroidal decomposition in a finite cylinder. I: Influence matrices for the magnetohydrodynamic equations

“…Instead, the main difficulty is that the magnetic field is not specified at the domain boundary but must instead satisfy matching conditions between the interior domain (here a finite cylinder) and the exterior domain (here an infinite vacuum). We have developed a formalism involving the Dirichlet-to-Neumann mapping for eliminating the exterior magnetic field, which has been implemented and validated for a two-dimensional test problem [29]. Future research will focus on implementing the poloidal-toroidal formulation for the full magnetohydrodynamic problem.…”

“…This approach is discussed and shown to perform quite well for a two-dimensional test problem in [29]. The exterior fields generated approximate the solution uniformly near the boundary and converge exponentially with resolution; the influence matrix representing the Dirichlet-to-Neumann mapping is well conditioned.…”