High-order finite-difference methods for Poisson’s equation

TLDR: In this paper, a finite-difference approximation to the three boundary value problems for Poisson's equation is given, with discretization errors of O(H^3) for the mixed boundary value problem, O(h^3 |ln(h)| for the Neumann problem, and O( h^4 ) for the Dirichlet problem respectively.

Abstract: In this thesis finite-difference approximations to the three boundary value problems for Poisson’s equation are given, with discretization errors of O(H^3) for the mixed boundary value problem, O(H^3 |ln(h)| for the Neumann problem and O(H^4)for the Dirichlet problem respectively . First an operator is constructed, which approximates the nor-ma1 derivative with a truncation errors of O(H^3). The derivation by which this result is obtained contains an improvement upon the one used for a similar operator by Bramble and Hubbard; it became thus possible to make their results valid under more general conditions. For points in a square net where the nine-point approximation to the Laplace operator cannot be used, because of their position near the boundary of the region under consideration, several approximations to the Laplace operator are given, dependent on the particular point configuration, which are all O(H^2). The above-mentioned operators are then used to formulate finite-difference problems with solutions approximating the corresponding continuous problems with the desired accuracy. These error bounds are an improvement of O(H) upon the most accurate approximations known for the Neumann and Robin problems, while the approximation given for the Dirichlet problem has as an advantage over a similar approximation given by Bramble and Hubbard, which is of the same order, that its res-ulting coefficient matrix is of positive type. All our approximations share this important property. For all three problems simple numerical examples are given. It is also pointed out that certain results of Bramble and Hubbard, with respect to the error in the numerical partial derivatives of the solutions, are valid for our approximations.