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https://doi.org/10.1006/jcph.1996.0237
Copy DOIJournal: Journal of Computational Physics | Publication Date: Nov 1, 1996 |
Citations: 6 |
We consider Laplace's equation in three dimensions where the domain is restricted to a finite region with the introduction of an artificial boundaryBon which a boundary condition is imposed. The finite difference method is employed to compare the solution at the nodes inside and on the surfaceBfor four different boundary conditions of which two are local and two are nonlocal. The standard nonlocal (DtN) boundary condition is derived from the solution of the exterior Dirichlet problem, and a discretized (DDtN) version is derived that applies at the nodes onB. However, the coefficients associated with the nodes onBin the system of linear equations for the solution is not sparse. This lack of sparsity is acute for three-dimensional problems owing to the large number of equations. The DDtN boundary condition is approximated to obtain a sparse nonlocal boundary condition, where the coefficients associated with the nodes onBare relatively sparse. We show that the DDtN solution is very accurate. In addition, we present results which indicate that the difference between the DDtN solution and the solution for each of the other three boundary conditions has the correct behavior when the artificial boundary is enlarged.
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