Iterative realization of finite difference schemes in the fictitious domain method for elliptic problems with mixed derivatives

  • Vasily M. Volkov Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus
  • Alena V. Prakonina Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus
DOI: https://doi.org/10.33581/2520-6508-2019-1-69-76

Abstract

Development of efficient finite difference schemes and iterative methods for solving anisotropic diffusion problems in an arbitrary geometry domain is considered. To simplify the formulation of the Neumann boundary conditions, the method of fictitious domains is used. On the example of a two-dimensional model problem of potential distribution in an isolated anisotropic ring conductor a comparative efficiency analysis of some promising finite-difference schemes and iterative methods in terms of their compatibility with the fictitious domain method is carried out. On the basis of numerical experiments empirical estimates of the asymptotic dependence of the convergence rate of the biconjugate gradient method with Fourier – Jacobi and incomplete LU factorization preconditioners on the step size and the value of the small parameter determining the continuation of the conductivity coefficient in the fictitious domain method are obtained. It is shown, that for one of the considered schemes the Fourier – Jacobi preconditioner is spectrally optimal and allows to eliminate the asymptotical dependence of the iterations number to achieve a given accuracy both on the value of the step size and the value of the small parameter in the fictitious domain method.

Author Biographies

Vasily M. Volkov, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

doctor of science (physics and mathematics); professor at the department of web-technologies and computer modeling, faculty of mechanics and mathematics

Alena V. Prakonina, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

senior lecturer at the department of web-technologies and computer modeling, faculty of mechanics and mathematics

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Published
2019-04-08
Keywords: finite-difference schemes, elliptic equations, mixed derivatives, iterative methods, fictitious domain method
How to Cite
Volkov, V. M., & Prakonina, A. V. (2019). Iterative realization of finite difference schemes in the fictitious domain method for elliptic problems with mixed derivatives. Journal of the Belarusian State University. Mathematics and Informatics, 1, 69-76. https://doi.org/10.33581/2520-6508-2019-1-69-76
Issue
No 1 (2019): Journal of the Belarusian State University. Mathematics and Informatics
Section
Computational Mathematics

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