Abstract
A boundary value problem for Maxwell’s equations in the frequency domain with impedance boundary conditions is considered. The problem is reduced to solving a system of two boundary integral equations containing weakly and strongly singular integrals. For the numerical solution of a system of integral equations, the paper presents a numerical solution method based on piecewise constant approximation and collocation methods. Thus, the original problem is reduced to solving a system of linear algebraic equations with a dense matrix. To effectively solve a system of linear equations, the method of mosaic-skeleton approximations of matrices is used. The specifics of applying the method of mosaic-skeleton approximations in this problem are analyzed.
This is a preview of subscription content, log in via an institution to check access.
REFERENCES
M. A. Leontovich, Investigations on Radiowave Propagation, Part II (Akad. Nauk, Moscow, 1948) [in Russian].
T. B. A. Senior, ‘‘A note on impedance boundary conditions,’’ Canad. J. Phys. 40, 663–665 (1962).
D. J. Hoppe and Y. Rahmat-Samii, Impedance Boundary Conditions in Electromagnetics (Taylor and Francis, Washington DC, 1995).
E. V. Zakharov and A. V. Setukha, ‘‘Method of boundary integral equations in the problem of diffraction of a monochromatic electromagnetic wave by a system of perfectly conducting and piecewise homogeneous dielectric objects,’’ Differ. Equat. 56, 1153–1166 (2020).
A. Setukha and S. Fetisov, ‘‘The method of relocation of boundary condition for the problem of electromagnetic wave scattering by perfectly conducting thin objects,’’ J. Comput. Phys. 373, 631–647 (2018).
E. Tyrtyshnikov, ‘‘Incomplete cross approximation in the mosaic-skeleton method,’’ Computing 64, 367–380 (2000).
A. Aparinov, A. Setukha, and S. Stavtsev, ‘‘Parallel implementation for some applications of integral equations method,’’ Lobachevskii J. Math. 39, 477–485 (2018).
J. L. Volakis and K. Sertel, Integral Equation Methods for Electromagnetic (SciTech, Raleigh, NC, 2012).
W. Gibson, The Method of Moments in Electromagnetics (Chapman and Hall/CRC, Boca Raton, 2008).
E. V. Zakharov, G. V. Ryzhakov, and A. V. Setukha, ‘‘Numerical solution of 3D problems of electromagnetic wave diffraction on a system of ideally conducting surfaces by the method of hypersingular integral equations,’’ Differ. Equat. 50, 1240–1251 (2014).
D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (SIAM, Philadelphia, 2013).
A. Setukha and E. Bezobrazova, ‘‘The method of hypersingularintegral equations in the problem of electromagnetic wave diffraction by a dielectric body with a partial perfectly conducting coating,’’ Russ. J. Numer. Anal. Math. Model. 32, 371–380 (2017).
A. A. Aparinov, A. V. Setukha, and S. L. Stavtsev, ‘‘Low rank methods of approximations in an electromagnetic problem,’’ Lobachevskii J. Math. 40, 1771–1780 (2019).
Funding
This study was conducted within the scientific program of the National Center of Physics and Mathematics, section no. 2—‘‘Mathematical Modeling on Zetta-scale and Exa-scale Supercomputers. Stage 2023-2025.’’ Also the work was supported by the Russian Science Foundation project no. 19-11-00338, https://rscf.ru/en/project/19-11-00338/, and by the Moscow Center of Fundamental and Applied Mathematics at INM RAS (Agreement with the Ministry of Education and Science of the Russian Federation no. 075-15- 2022-286).
Author information
Authors and Affiliations
Corresponding authors
Additional information
(Submitted by N. B. Pleshchinskii)
Rights and permissions
About this article
Cite this article
Setukha, A.V., Stavtsev, S.L. On the Application of Mosaic-Skeleton Approximations of Matrices in Electrodynamics Problems with Impedance Boundary Conditions. Lobachevskii J Math 44, 4062–4069 (2023). https://doi.org/10.1134/S199508022309038X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S199508022309038X