On the optimal discretization of the solution Poisson’s equation

Authors

DOI:

https://doi.org/10.26577/JMMCS2024-122-02-b6
        129 76

Keywords:

Poisson's equation, discretization operator, optimal discretization, Fourier coefficients, discretization error, linear functionals, Sobolev class

Abstract

The paper studies the problem of discretizing the solution of the Poisson equation with the right hand side f  belonging to the multidimensional periodic Sobolev class. The research methodology is based on considering the problem of discretizing the solution of the Poisson equation as one of the concretizations of the general problem of optimal recovery of the operator Tf and using well known statements of approximation theory. Within the framework of this general optimal recovery problem, we first estimate from above the smallest discretization error N of the solution of the Poisson equation in the Hilbert metric using the discretization operator (l(N) , N) constructed from a finite set of Fourier coefficients of the function f. A lower estimate, coinciding in order with the upper estimate, for the smallest error N was obtained by involving all linear functionals defined on the multidimensional Sobolev class. It should be noted that the optimal discretization operator (l(N) , N) better approximates the solution under consideration in the Hilbert metric than any discretization operator constructed from values f at given points. Poisson’s equation is an elliptic partial differential equation and describes many physical phenomena such as electrostatic field, stationary temperature field, pressure field and velocity potential field in hydrodynamics. Therefore, the relevance of the research conducted here is beyond doubt.

References

Korobov N.M., Teoretiko– chislovye metody v priblizhennom analize [Numerical– theoretic methods in approximate

analysis].- M., 1963. [in Russian]

Bailov, E.A., Temirgaliev, N., "Discretization of the Solutions to Poisson’s Equation", Computational Mathematics and Mathematical physics 46, no.9 (2006): 1515– 1525.

Kudaibergenov, S.S., Sabitova, S.G, "Discretization of solutions to Poisson’s equation in the Korobov class", Computational Mathematics and Mathematical physics 53, no.7 (2013): 896– 907.

Sickel, W. and Ullrich, T., "The Smolyak’s algorithm, sampling on sparse grids and function spaces of dominating mixed smoothness", East J. Approx. 73, 193, no. 4 (2007): 287- 425.

Naurizbayev N., Temirgaliyev N., "An exact order of discrepancy of the Smolyak grid and some general conclusionc in the theory of numerical integrations", Found Comput Math 12 (2012): : 139- 172.

Temirgaliev, N., Kudaibergenov, S.S., and Shomanova, A.A., "An application of tensor products of functional in problems of numerical integration", Izv. Math. 73, no.2 (2009): 393- 434.

Utesov, A.B., "On Error Estimates for Discretization Operators for the Solution of the Poisson Equation", Differential Equations 60, no. 1 (2024): 136– 143.

Arystangalikyzy, A., "Discretization of solutions of Poisson equation by inaccurate information", Bulletin of the L.N.Gumilov Eurasian National University. Mathematics. Computer Science. Mechanics Series 144, no.3 (2023): 39– 44.

Bailov E.A., Priblizhennoe integrirovanie i vosstanovlenie funkcij iz anizotropnyh klassov i vosstanovlenie reshenij uravnenija Puassona [Approximate integration and restoration of functions from anisotropic classes and restoration of solutions to the Poisson equation]: diss. ... kand. fiz.- mat. nauk. Almaty. (1998) [In Russian]

Utesov, A.B., Bazarkhanova, A.A., "On Optimal Discretization of Solutions of the Heat Equation and the Limit Error of the Optimum Computing Unit", Differential Equations 57, no.12 (2021): 1726–1735.

Temirgaliev, N., "Teoretiko– chislovye metody i teoretiko– verojatnostnyj podhod k zadacham analiza. Teorija vlozhenij i priblizhenij, absoljutnaja shodimost’ i preobrazovanija rjadov Fur’e [Numerical–theoretical methods and a probability– theoretical approach to the problems of analysis.The theory of embeddings and approximations, absolute convergence and transformations of Fourier series]", Vestnik Evrazijskogo universiteta no.3, (1997): 90– 144. [in Russian].

Utesov, A.B., "Optimal Recovery of Functions from Numerical Information on Them and Limiting Error of the Optimal Computing Unit", Mathematical Notes 11, no.5 (2022): 759– 769.

Azhgaliev, Sh. U., "Discretization of the solutions of the heat equation", Math. Notes 82, no.2 (2007): 153– 158.

Temlyakov V., Multivariate Approximation.- Cambridge University Press, 2018.- 551 p

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How to Cite

Utesov, . A. ., & Shanauov, R. . (2024). On the optimal discretization of the solution Poisson’s equation. Journal of Mathematics, Mechanics and Computer Science, 122(2), 67–76. https://doi.org/10.26577/JMMCS2024-122-02-b6