A fourth order accurate approximation of the solution of Laplace's equation on a rectangle using the two-stage difference method
CitationSarıkaya, H. (2021). A fourth order accurate approximation of the solution of Laplace's equation on a rectangle using the two-stage difference method. Fourth International Conference of Mathematical Sciences, Maltepe Üniversitesi. s. 1-5.
In this paper, two stage difference method is presented to solve the Dirichlet problem for the Laplace equation on rectangle. In the first stage, the sum of the pure fourth order derivatives of the required solution is approximated on a square grid. Then, by using the quantities that are determined in the first stage, the system of difference equations which approximates the Dirichlet problem, is computed during the second stage. The difference equations found in the stages are formulated by using the 5−point averaging operator. Due to these facts that, the boundary values are continuous and sixth times differentiable at the edges of the rectangle, the derivatives of them satisfy Holder ¨ condition and at the end, their second and fourth order derivatives meet the matching condition implied by the Laplace equation. We proved that the difference solution of the Dirichlet problem is uniform convergent with the order O(h4), where h denotes the mesh size.
SourceFourth International Conference of Mathematical Sciences
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