Skip to main content
SpringerLink
Log in

Consistent convergence rate estimates in the grid W 2,0 2 (ω) norm for difference schemes approximating nonlinear elliptic equations with mixed derivatives and solutions from W 2,0 m (Ω), 3 < m ≤ 4

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

The Dirichlet boundary value problem for nonlinear elliptic equations with mixed derivatives and unbounded nonlinearity is considered. A difference scheme for solving this class of problems and an implementing iterative process are constructed and investigated. The convergence of the iterative process is rigorously analyzed. This process is used to prove the existence and uniqueness of a solution to the nonlinear difference scheme approximating the original differential problem. Consistent with the smoothness of the desired solution, convergence rate estimates in the discrete norm of W 2,0 2 (ω) for difference schemes approximating the nonlinear equation with unbounded nonlinearity are established.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. M. Karchevskii and A. D. Lyashko, Finite Difference Schemes for Nonlinear Partial Differential Equations (Kazan. Gos. Univ., Kazan, 1976) [in Russian].

    Google Scholar 

  2. A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1989; Marcel Dekker, New York, 2001).

    MATH  Google Scholar 

  3. A. A. Samarskii and V. B. Andreev, Difference Schemes for Elliptic Equations (Nauka, Moscow, 1976) [in Russian].

    MATH  Google Scholar 

  4. A. A. Samarskii and P. N. Vabishchevich, Computational Heat Transfer (Wiley, New York, 1996; Librokom, Moscow, 2009).

    Google Scholar 

  5. A. A. Samarskii, R. D. Lazarov, and V. L. Makarov, Difference Schemes for Differential Equations with Weak Solutions (Vysshaya Shkola, Moscow, 1987) [in Russian].

    Google Scholar 

  6. A. A. Samarskii, “Investigation of the accuracy of difference schemes for problems with weak solutions,” in Topical Problems in Mathematical Physics and Numerical Mathematics (Nauka, Moscow, 1984), pp. 174–183 [in Russian].

    Google Scholar 

  7. P. P. Matus, “Two-level difference schemes with variable weights,” Vestn. Akad. Nauk Belarusi Ser. Fiz.-Mat. Nauk, No. 4, 15–21 (1993).

    MATH  Google Scholar 

  8. L. V. Maslovskaya, “On the convergence of difference methods for certain degenerate quasi-linear parabolic equations,” USSR Comput. Math. Math. Phys. 12 (6), 91–105 (1972).

    Article  MATH  Google Scholar 

  9. V. N. Abrashin, “Difference schemes for nonlinear hyperbolic equations 2,” Differ. Uravn. 11 (2), 294–308 (1975).

    MathSciNet  MATH  Google Scholar 

  10. V. N. Abrashin, Doctoral Dissertation in Mathematics and Physics (Vychisl. Tsentr Akad. Nauk SSSR, Moscow, 1979).

    Google Scholar 

  11. V. N. Abrashin and V. A. Asmolik, “Locally one-dimensional difference schemes for multidimensional quasilinear hyperbolic equations,” Differ. Uravn. 18 (7), 1107–1117 (1982).

    MathSciNet  MATH  Google Scholar 

  12. A. D. Lyashko and E. M. Fedotov, “Investigation of nonlinear two-level operator-difference schemes with weights,” Differ. Uravn. 21 (7), 1217–1227 (1985).

    MATH  Google Scholar 

  13. P. P. Matus, “Unconditional convergence of some difference schemes for gas dynamics problems,” Differ. Uravn. 21 (7), 1227–1238 (1985).

    Google Scholar 

  14. P. P. Matus and L. V. Stanishevskaya, “Unconditional convergence of difference schemes for nonstationary quasilinear equations of mathematical physics,” Differ. Equations 27 (7), 847–859 (1991).

    MathSciNet  MATH  Google Scholar 

  15. P. P. Matus, M. N. Moskal’kov, and V. S. Shcheglik, “Consistent estimates for rate of convergence of the grid method for a second-order nonlinear equation with generalized solutions,” Differ. Equations 31 (7), 1198–1206 (1995).

    MathSciNet  MATH  Google Scholar 

  16. V. S. Shcheglik, “An analysis of a difference scheme approximating a third boundary-value problem for a second-order nonlinear differential equation,” Comput. Math. Math. Phys. 37 (8), 919–925 (1997).

    MathSciNet  Google Scholar 

  17. B. S. Jovanović and E. Süli, Analysis of Finite Difference Schemes (Springer Science & Business Media, London, 2014).

    Book  MATH  Google Scholar 

  18. B. S. Jovanović, “Finite difference scheme for partial differential equations with weak solutions and irregular coefficients,” Comput. Methods Appl. Math. 4 (1), 48–65 (2004).

    MathSciNet  MATH  Google Scholar 

  19. G. Berikelashvili, “On a nonlocal boundary-value problem for two-dimensional elliptic equation,” Comput. Methods Appl. Math. 3 (1), 35–44 (2003).

    MathSciNet  MATH  Google Scholar 

  20. G. Berikelashvili, M. M. Gupta, and M. Mirianashvili, “Convergence of fourth order compact difference schemes for three-dimensional convection–diffusion equations,” SIAM J. Numer. Anal. 45 (1), 443–455 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  21. G. Berikelashvili and M. Mirianashvili, “On the convergence of difference schemes for generalized Benjamin–Bona–Mahony equation,” Numer. Methods Partial Differ. Equations 30 (1), 301–320 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  22. B. S. Jovanović, P. P. Matus, and V. S. Shchehlik, “The estimates of accuracy of difference schemes for the nonlinear heat equation with weak solutions,” Math. Model. Anal. 5, 86–96 (2000).

    MathSciNet  MATH  Google Scholar 

  23. P. P. Matus, “Well-posedness of difference schemes for semilinear parabolic equations with weak solutions,” Comput. Math. Math. Phys. 50 (12), 2044–2063 (2010).

    Article  MathSciNet  Google Scholar 

  24. P. Matus, “On convergence of difference schemes for IBVP for quasilinear parabolic equations with generalized solutions,” Comput. Meth. Appl. Math. 14 (3), 361–371 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  25. F. P. Vasil’ev, Optimization Methods (Faktorial, Moscow, 2002) [in Russian].

    Google Scholar 

  26. M. M. Potapov, Approximation of Optimization Problems in Mathematical Physics (Hyperbolic Equations) (Mosk. Gos. Univ., Moscow, 1985) [in Russian].

    Google Scholar 

  27. A. Z. Ishmukhametov, Stability and Approximation of Optimal Control Problems (Vychisl. Tsentr Ross. Akad. Nauk, Moscow, 1999) [in Russian].

    MATH  Google Scholar 

  28. A. Z. Ishmukhametov, Stability and Approximation of Optimal Control Problems for Distributed Parameter Systems (Vychisl. Tsentr Ross. Akad. Nauk, Moscow, 2001) [in Russian].

    Google Scholar 

  29. F. V. Lubyshev, Difference Approximations of Optimal Control Problems for Systems Governed by Partial Differential Equations (Bashkir. Gos. Univ., Ufa, 1999) [in Russian].

    Google Scholar 

  30. F. V. Lubyshev and M. E. Fairuzov, “Approximations of optimal control problems for semilinear elliptic equations with discontinuous coefficients and states and with controls in the coefficients multiplying the highest derivatives,” Comput. Math. Math. Phys. 56 (7), 1238–1263 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  31. O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations (Nauka, Moscow, 1973; Academic, New York, 1987).

    Google Scholar 

  32. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics (Sib. Otd. Akad. Nauk SSSR, Novosibirsk, 1962; Am. Math. Soc., Providence, R.I., 1963).

    Google Scholar 

  33. O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Nauka, Moscow, 1973; Springer-Verlag, New York, 1985).

    Google Scholar 

  34. Linear Equations of Mathematical Physics (Holt, Rinehart and Winston, New York, 1967; Vysshaya Shkola, Moscow, 1977).

Download references

Author information

Authors and Affiliations

  1. Bashkir State University, Ufa, Bashkortostan, 450074, Russia

    F. V. Lubyshev & M. E. Fairuzov

Authors
  1. F. V. Lubyshev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. E. Fairuzov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. E. Fairuzov.

Additional information

Original Russian Text © F.V. Lubyshev, M.E. Fairuzov, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 9, pp. 1444–1470.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lubyshev, F.V., Fairuzov, M.E. Consistent convergence rate estimates in the grid W 2,0 2 (ω) norm for difference schemes approximating nonlinear elliptic equations with mixed derivatives and solutions from W 2,0 m (Ω), 3 < m ≤ 4. Comput. Math. and Math. Phys. 57, 1427–1452 (2017). https://doi.org/10.1134/S0965542517090081

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542517090081

Keywords

Search

Navigation