Numerical recipes: the art of scientific computing
Numerical recipes: the art of scientific computing
Divergence-free velocity fields in nonperiodic geometries
Journal of Computational Physics
Solution of Helmholtz equation in the exterior domain by elementary boundary integral methods
Journal of Computational Physics
A New Spectral Boundary Integral Collocation Method for Three-Dimensional Potential Problems
SIAM Journal on Numerical Analysis
Spectral methods in MatLab
A combination of spectral and finite elements for an exterior problem in the plane
Applied Numerical Mathematics
The integral equation method for a steady kinematic dynamo problem
Journal of Computational Physics
Journal of Computational Physics
Fast solvers for 3D Poisson equations involving interfaces in a finite or the infinite domain
Journal of Computational and Applied Mathematics
An interior penalty Galerkin method for the MHD equations in heterogeneous domains
Journal of Computational Physics
Journal of Computational Physics
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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.