For citation:
Dosiyev A. A. A highly accurate difference method for solving the Dirichlet problem of the Laplace equation on a rectangular parallelepiped with boundary values in C^(k,1). Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2024, vol. 24, iss. 2, pp. 162-172. DOI: 10.18500/1816-9791-2024-24-2-162-172, EDN: WNJQOX
A highly accurate difference method for solving the Dirichlet problem of the Laplace equation on a rectangular parallelepiped with boundary values in C^(k,1)
A three-stage difference method for solving the Dirichlet problem of Laplace's equation on a rectangular parallelepiped is proposed and justified. In the first stage, approximate values of the sum of the pure fourth derivatives of the solution are defined on a cubic grid by the 14-point difference operator. In the second stage, approximate values of the sum of the pure sixth derivatives of the solution are defined on a cubic grid by the simplest $6$-point difference operator. In the third stage, the system of difference equations for the sought solution is constructed again by using the $6$-point difference operator with the correction by the quantities determined in the first and the second stages. It is proved that the proposed difference solution to the Dirichlet problem converges uniformly with the order $O(h^{6}(|\ln h|+1))$, when the boundary functions on the faces are from $C^{7,1}$ and on the edges their second, fourth, and sixth derivatives satisfy the compatibility conditions, which follows from the Laplace equation. A numerical experiment is illustrated to support the analysis made.
- Tarjan R. E. Graph theory and Gaussian elimination. In: Bunch J. R., Rose D. J. (eds.) Sparse Matrix Computations. Academic Press, 1976, pp. 3–22. https://doi.org/10.1016/B978-0-12-141050-6.50006-4
- Fox L. Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations. Proceedings of the Royal Society A, 1947, vol. 190, pp. 31–59. https://doi.org/10.1098/rspa.1947.0060
- Woods L. C. Improvements to the accuracy of arithmetical solutions to certain two-dimensional field problems. The Quarterly Journal of Mechanics and Applied Mathematics, 1950, vol. 3, iss. 3, pp. 349–363. https://doi.org/10.1093/qjmam/3.3.349
- Volkov E. A. Solving the Dirichlet problem by a method of corrections with higher order differences. I. Differentsial’nye Uravneniya, 1965, vol. 1, iss. 7, pp. 946–960 (in Russian).
- Volkov E. A. Solving the Dirichlet problem by a method of corrections with higher order differences. II. Differentsial’nye Uravneniya, 1965, vol. 1, iss. 8, pp. 1070–1084 (in Russian).
- Berikelashvili G. K., Midodashvili B. G. Compatible convergence estimates in the method of refinement by higher-order differences. Differential Equations, 2015, vol. 51, iss. 1, pp. 107–115. https://doi.org/10.1134/S0012266115010103
- Volkov E. A. A two-stage difference method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped. Computational Mathematics and Mathematical Physics, 2009, vol. 49, iss. 3, pp. 496–501. https://doi.org/10.1134/S0965542509030117
- Volkov E. A., Dosiyev A. A. A highly accurate homogeneous scheme for solving the Laplace equation on a rectangular parallelepiped with boundary values in Ck1 . Computational Mathematics and Mathematical Physics, 2012, vol. 52, iss. 6, pp. 879–886. https://doi.org/10.1134/S0965542512060152
- Volkov E. A. On differential properties of solutions of the Laplace and Poisson equations on a parallelepiped and efficient error estimates of the method of nets. Proceedings of the Steklov Institute of Mathematics, 1969, vol. 105, pp. 54–78.
- Mikhailov V. P. Partial Differential Equations. Moscow, Mir, 1978. 396 p. (in Russian).
- Samarskii A. A. The Theory of Difference Schemes. Marcel, Dekker Inc., 2001. 761 p.
- Volkov E. A. Application of a 14-point averaging operator in the grid method. Computational Mathematics and Mathematical Physics, 2010, vol. 50, iss. 12, pp. 2023–2032. https://doi.org/10.1134/S0965542510120055
- Smith B., Bjorstad P., Gropp W. Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, 1996. 238 p.
- Volkov E. A. On the smoothness of solutions to the Dirichlet problem and the method of composite grids on polyhedra. Proceedings of the Steklov Institute of Mathematics, 1981, vol. 150, pp. 71–103.
- Volkov E. A. A method of composite grids on a prism with an arbitrary polygonal base. Proceedings of the Steklov Institute of Mathematics, 2003, vol. 243, pp. 1–23. https://doi.org/10.1046/j.1468-2982.2003.00539.x
- Volkov E. A. On the grid method for approximating the derivatives of the solution of the Dirichlet problem for the Laplace equation on the rectangular parallelepiped. Russian Journal of Numerical Analysis and Mathematical Modelling, 2004, vol. 19, iss. 3, pp. 269–278. https://doi.org/10.1515/1569398041126500
- Dosiyev A. A. The high accurate block-grid method for solving Laplace’s boundary value problem with singularities. SIAM Journal on Numerical Analysis, 2004, vol. 42, iss. 1, pp. 153–178. https://doi.org/10.1137/S0036142900382715
- Dosiyev A. A. The block-grid method for the approximation of the pure second order derivatives for the solution of Laplace’s equation on a staircase polygon. Journal of Computational and Applied Mathematics, 2014, vol. 259, pt. A, pp. 14–23. https://doi.org/10.1016/j.cam.2013.03.022
- Dosiyev A. A., Sarikaya H. A highly accurate difference method for approximating the solution and its first derivatives of the Dirichlet problem for Laplace’s equation on a rectangle. Mediterranean Journal of Mathematics, 2021, vol. 18, art. 252. https://doi.org/10.1007/s00009-021-01900-8
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