Advertisement

Journal of Mathematical Sciences

, Volume 159, Issue 2, pp 229–240 | Cite as

Estimates of deviations from exact solutions of initial boundary value problems for the wave equation

  • S. Repin
Article

We derive computable upper bounds of the difference between an exact solution of the initial boundary value problem for a linear hyperbolic equation and any function in a space-time cylinder that belongs to the respective energy class. We prove that the bounds vanish if and only if the approximate solution coincides with the exact one. Bibliography: 13 titles.

Keywords

Variational Inequality Steklov Institute Posteriori Error Initial Boundary Parabolic Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer, New York (1985).MATHGoogle Scholar
  2. 2.
    L. Evans, Partial Differential Equations, Am. Math. Soc., Providence, RI (1998).MATHGoogle Scholar
  3. 3.
    S. Repin, “A posteriori error estimation for nonlinear variational problems by duality theory” [in Russian], Zap. Nauchn. Semin. POMI 243, 201-214 (1997).Google Scholar
  4. 4.
    S. Repin, “A posteriori error estimation for variational problems with uniformly convex functionals,” Math. Comput. 69, 481–500 (2000).MATHMathSciNetGoogle Scholar
  5. 5.
    S. Repin, “Two-sided estimates of deviation from exact solutions of uniformly elliptic equations” [in Russian], Trudy S. Peterb. Mat. Ovschv. 9, 143-171 (2001); English transl.: Am. Math. Soc. Transl. Ser. 2 209, Am. Math. Soc., Providence, RI (2003).MathSciNetGoogle Scholar
  6. 6.
    S. Repin, “Estimates of deviation from exact solutions of initial boundary value problems for the heat equation,” Rend. Mat. Acc. Lincei, 13, 121-133 (2002).MATHMathSciNetGoogle Scholar
  7. 7.
    A. Gaevskaya and S. Repin, “A posteriori error estimates for approximate solutions of linear parabolic problems” [In Russian], Different. Equations 41, No. 7, 970-983 (2005); English transl.: Differ. Equ. 41, No. 7 (2005).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    S. Repin, “Estimates of deviations from exact solutions of elliptic variational inequalities”[in Russian], Zap. Nauchn. Semin. POMI 271, 188-203 (2000).Google Scholar
  9. 9.
    M. Bildhauer, M. Fuchs, and S. Repin, “Duality based a posteriori error estimates for higher order variational inequalities with power growth functionals,” Ann. Acad. Scent. Fenn., Math. 33, 475-490 (2008).MATHMathSciNetGoogle Scholar
  10. 10.
    S. Repin, “Estimates of deviations from exact solutions of elliptic variational inequalities” [in Russian], Zap. Nauchn, Semin. POMI 326, 147-164 (2007).Google Scholar
  11. 11.
    S. Repin, A Posteriori Error Estimates for Partial Differential Equations, Walter de-Gruyter, Berlin (2008).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.St. Petersburg Department of V.A. Steklov Institute of MathematicsSt. PetersburgRussia

Personalised recommendations