Advertisement

Lobachevskii Journal of Mathematics

, Volume 39, Issue 9, pp 1466–1477 | Cite as

High-Order Accuracy Approximation for a Two-Point Boundary Value Problem of Fourth Order with Degenerate Coefficients

  • A. A. SobolevEmail author
  • M. R. Timerbaev
Selected Articles from the Journal Uchenye Zapiski Kazanskogo Universiteta, Seriya Fiziko-Matematicheskie Nauki

Abstract

High-order accurate finite element schemes for a fourth-order ordinary differential equation with degenerate coefficients on the boundary are constructed. The method for solving the problem is based on multiplicative and additive-multiplicative separation of singularities. For right-hand sides of the given class of smoothness, an optimal convergence rate is proved.

Keywords and phrases

two-point boundary value problem finite element schemes weighted function spaces multiplicative and additive-multiplicative separation of singularity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Sh. I. Tayupov and M. R. Timerbaev, “Finite element schemes of a high accuracy order for two-pointed heterogeneous boundary-value problem with degeneration,” Uchen. Zap. Kazan. Univ., Ser. Fiz.-Mat. Nauki 148 (4), 63–75 (2006).zbMATHGoogle Scholar
  2. 2.
    A. A. Sobolev and M. R. Timerbaev, “On finite element method of high-order accuracy for two-point degenerated Dirichlet problem of 4th order,” Uchen. Zap. Kazan. Univ., Ser. Fiz.-Mat. Nauki 152 (1), 235–244 (2010).zbMATHGoogle Scholar
  3. 3.
    M. R. Timerbaev, “Multiplicative extraction of singularities in FEM solvers for degenerate elliptic equations,” Differ. Equations 36 (7), 1086–1093 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    L. D. Kudryavtsev, “Equivalent norms in weighted spaces,” Proc. Steklov Inst. Math. 170 (10), 185–218 (1987).zbMATHGoogle Scholar
  5. 5.
    S. M. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems (Nauka, Moscow, 1977; Springer, Berlin, Heidelberg, 1975).Google Scholar
  6. 6.
    H. Tribal, Interpolation Theory, Function Spaces, Differential Operators (North-Holland, Amsterdam, 1978).Google Scholar
  7. 7.
    M. R. Timerbaev, “Weighted estimates for the solution of the Dirichlet problem with anisotropic degeneration on part of the boundary,” Russ. Math. (Iz. VUZ,Mat.) 47, 58–71 (2003).MathSciNetzbMATHGoogle Scholar
  8. 8.
    M. R. Timerbaev, “On FEM schemes for a 2-point boundary value Dirichlet problem of the fourth order with weak degeneracy,” Issled. Prikl. Mat. Inform., No. 25, 78–85 (2004).Google Scholar
  9. 9.
    Ph. Ciarlet, The Finite Element Method for Elliptic Problems (SIAM, Philadelphia, 2002).CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Kazan (Volga Region) Federal UniversityKazanRussia

Personalised recommendations