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 Zh. Vychisl. Mat. Mat. Fiz., 2009, Volume 49, Number 3, Pages 512–517 (Mi zvmmf27)

A two-stage difference method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped

E. A. Volkov

Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119333, Russia

Abstract: A novel two-stage difference method is proposed for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped. At the first stage, approximate values of the sum of the pure fourth derivatives of the desired solution are sought on a cubic grid. At the second stage, the system of difference equations approximating the Dirichlet problem is corrected by introducing the quantities determined at the first stage. The difference equations at the first and second stages are formulated using the simplest six-point averaging operator. Under the assumptions that the given boundary values are six times differentiable at the faces of the parallelepiped, those derivatives satisfy the Hölder condition, and the boundary values are continuous at the edges and their second derivatives satisfy a matching condition implied by the Laplace equation, it is proved that the difference solution to the Dirichlet problem converges uniformly as $O(h^4\ln h^{-1})$, where $h$ is the mesh size.

Key words: numerical solution to the Laplace equation, convergence of difference solutions, domain in the form of a rectangular parallelepiped.

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English version:
Computational Mathematics and Mathematical Physics, 2009, 49:3, 496–501

Bibliographic databases:

UDC: 519.632.4

Citation: E. A. Volkov, “A two-stage difference method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped”, Zh. Vychisl. Mat. Mat. Fiz., 49:3 (2009), 512–517; Comput. Math. Math. Phys., 49:3 (2009), 496–501

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Berikelashvili G.K. Midodashvili B.G., “Compatible Convergence Estimates in the Method of Refinement By Higher-Order Differences”, Differ. Equ., 51:1 (2015), 107–115
2. Berikelashvili G., Midodashvili B., “on Increasing the Convergence Rate of Difference Solution To the Third Boundary Value Problem of Elasticity Theory”, Bound. Value Probl., 2015, 226
3. Berikelashvili G., Gupta M.M., Midodashvili B., “on the Improvement of Convergence Rate of Difference Schemes With High Order Differences For a Convection-Diffusion Equation”, Proceedings of the International Conference of Numerical Analysis and Applied Mathematics 2014 (Icnaam-2014), AIP Conference Proceedings, 1648, eds. Simos T., Tsitouras C., Amer Inst Physics, 2015, UNSP 470002
4. Berikelashvili G. Midodashvili B., “Method of corrections by higher order differences for elliptic equations with variable coefficients”, Georgian Math. J., 23:2 (2016), 169–180
5. Berikelashvili G., Midodashvili B., “Method of Corrections By Higher Order Differences For Poisson Equation With Nonlocal Boundary Conditions”, Trans. A Razmadze Math. Inst., 170:2 (2016), 287–296
6. Dosiyev A.A., Sarikaya H., “A Highly Accurate Difference Method For Solving the Dirichlet Problem For Laplace'S Equation on a Rectangle”, International Conference Functional Analysis in Interdisciplinary Applications (FAIA2017), AIP Conference Proceedings, 1880, eds. Kalmenov T., Sadybekov M., Amer Inst Physics, 2017, UNSP 040006
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