Abstract
A three-dimensional scalar stationary scattering problem is considered. It is formulated in the form of a weakly singular Fredholm boundary integral equation of the first kind with a single unknown function. The equation is approximated by a system of linear algebraic equations, which is then solved numerically by an iterative method. The mosaic-skeleton method is used at the stage of the approximate solution of this system in order to reduce the computational complexity of the approach.
Similar content being viewed by others
REFERENCES
V. I. Dmitriev and E. V. Zakharov, Integral Equations in Boundary Value Problems of Electrodynamics (Mosk. Gos. Univ., Moscow, 1987) [in Russian].
M. Costabel and E. Stephan, “A direct boundary integral equation method for transmission problems,” J. Math. Anal. Appl. 106, 367–413 (1985).
D. Colton and R. Kress, Integral Equation Methods in ScatteringTheory (Wiley, New York, 1984).
V. A. Tsetsokho, V. V. Voronin, and S. I. Smagin, “On the solution of diffraction problems using single-layer potentials,” Dokl. Akad. Nauk SSSR 302 (2), 323–327 (1988).
R. E. Kleinman and P. A. Martin, “On single integral equations for the transmission problem of acoustics,” SIAM J. Appl. Math. 48 (2), 307–325 (1988).
S. I. Smagin, Doctoral Dissertation in Mathematics and Physics (Novosibirsk, 1991).
A. A. Kashirin, Candidate’s Dissertation in Mathematics and Physics (Khabarovsk, 2006).
A. A. Kashirin and S. I. Smagin, “Generalized solutions of the integral equations of a scalar diffraction problem,” Differ. Equations 42 (1), 88–100 (2006).
A. A. Kashirin and S. I. Smagin, “Numerical solution of integral equations for a scalar diffraction problem,” Dokl. Math. 90 (2), 549–552 (2014).
A. A. Kashirin and S. I. Smagin, “Numerical solution of integral equations for three-dimensional scalar diffraction problems,” Vychisl. Tekhnol. 23 (2), 20–36 (2018).
S. I. Smagin, “Numerical solution of an integral equation of the first kind with a weak singularity for the density of a single layer potential,” USSR Comput. Math. Math. Phys. 28 (6), 41–49 (1988).
A. A. Kashirin and S. I. Smagin, “Potential-based numerical solution of Dirichlet problems for the Helmholtz equation”, Comput. Math. Math. Phys. 52 (8), 1173–1185 (2012).
A. A. Kashirin, S. I. Smagin, and M. Yu. Taltykina, “Mosaic-skeleton method as applied to the numerical solution of three-dimensional Dirichlet problems for the Helmholtz equation in integral form,” Comput. Math. Math. Phys. 56 (4), 612–625 (2016).
A. A. Kashirin and M. Yu. Taltykina, “On the existence of mosaic-skeleton approximations for discrete analogues of integral operators,” Comput. Math. Math. Phys. 57 (9), 1404–1415 (2017).
Y. Saad and M. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7 (3), 856–869 (1986).
E. E. Tyrtyshnikov, “Fast multiplication methods and solving equations,” in Matrix Methods and Computations (Inst. Vychisl. Mat. Ross. Akad. Nauk, Moscow, 1999), pp. 4–41 [in Russian].
S. A. Goreinov, “Mosaic-skeleton approximations of matrices generated by asymptotically smooth and oscillatory kernels,” in Matrix Methods and Computations (Inst. Vychisl. Mat. Ross. Akad. Nauk, Moscow, 1999), pp. 42–76 [in Russian].
E. E. Tyrtyshnikov, “Incomplete cross approximations in the mosaic-skeleton method,” Computing 64 (4), 367–380 (2000).
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations (Cambridge Univ. Press, Cambridge, 2000).
D. V. Savost’yanov, Candidate’s Dissertation in Mathematics and Physics (Moscow, 2006).
C. D. Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, Philadelphia, PA, 2000).
A. A. Sorokin, S. V. Makogonov, and S. P. Korolev, “The information infrastructure for collective scientific work in the Far East of Russia,” Sci. Tech. Inf. Proc. 44 (4), 302–304 (2017).
A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1999; Dover, New York, 2011).
Funding
This work was supported by the Russian Foundation for Basic Research (project nos. 17-01-00682, 20-01-00450) and by the Far Eastern Branch of the Russian Academy of Sciences (basic research program, project no. 18-5-100).
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by I. Ruzanova
Rights and permissions
About this article
Cite this article
Kashirin, A.A., Smagin, S.I. & Timofeenko, M.Y. Parallel Mosaic-Skeleton Algorithm for the Numerical Solution of a Three-Dimensional Scalar Scattering Problem in Integral Form. Comput. Math. and Math. Phys. 60, 895–910 (2020). https://doi.org/10.1134/S0965542520050097
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542520050097