Abstract
Integral formulas are presented for approximating the surface gradient (of a scalar function given on a surface) and divergence (of a tangent vector field given on a surface) that are analogs of the well-known formulas for the derivatives of a function on a plane. Estimates of the error in the approximation of these functions are obtained. The question of subsequent approximation of the integrals that give expression for the surface gradient and divergence by quadrature sums over the values of the function under study at the nodes selected on the cells of the unstructured grid approximating the surface is also considered.
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REFERENCES
Colton, D. and Kress, R., Integral Equation Methods in Scattering Theory, New York–Chichester–Brisbane–Toronto–Singapore: Wiley-Interscience, 1983. Translated under the title: Metody integral’nykh uravnenii v teorii rasseyaniya, Moscow: Mir, 1987.
Lifanov, I.K., Metod singulyarnykh integral’nykh uravnenii i chislennyi eksperiment (Method of Singular Integral Equations and Numerical Experiment), Moscow: Yanus, 1995.
Volakis, J.L. and Sertel, K., Integral Equation Methods for Electromagnetics, Raleigh: Inst. Eng. Technol., 2012.
Pisarev, I.V. and Setukha, A.V., Transfer of the boundary condition to the median surface in the numerical solution of the boundary value problem of the linear wing theory, Vychisl. Metody Programm., 2014, vol. 15, no. 1, pp. 109–120.
Setukha, A. and Fetisov, S., The method of relocation of boundary condition for the problem of electromagnetic wave scattering by perfectly conducting thin objects, J. Comput. Phys., 2018, vol. 373, pp. 631–647.
Gutnikov, V.A., Lifanov, I.K., and Setukha, A.V., Simulation of the aerodynamics of buildings and structures by means of the closed vortex loop method, Fluid Dyn., 2006, vol. 41, pp. 555–567.
Eldredge, J.D., Leonard, A., and Colonius, T., A general deterministic treatment of derivatives in particle methods, J. Comput. Phys., 2002, vol. 180, pp. 686–709.
Zorich, V.A., Matematicheskii analiz. Ch. 1 (Mathematical Analysis. Part 1), Moscow: Fazis, 1997.
Zakharov, E.V., Ryzhakov, G.V., and Setukha, A.V., Numerical solution of 3d problems of electromagnetic wave diffraction on a system of ideally conducting surfaces by the method of hypersingular integral equations, Differ. Equations, 2014, vol. 50, no. 9, pp. 1240–1251.
Ryzhakov, G.V. and Setukha, A.V., Convergence of a numerical scheme of the type of the vortex loop method on a closed surface with an approximation to the surface shape, Differ. Equations, 2012, vol. 48, no. 9, pp. 1308–1317.
Funding
The work was supported by the Ministry of Science and Higher Education of the Russian Federation within the framework of the program of the Moscow Center for Fundamental and Applied Mathematics under agreement no. 075-15-2022-286.
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Translated by V. Potapchouck
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Setukha, A.V. On Approximation of Surface Derivatives of Functions with the Application of Integral Operators. Diff Equat 59, 847–861 (2023). https://doi.org/10.1134/S0012266123060125
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DOI: https://doi.org/10.1134/S0012266123060125