Numerische Mathematik

, Volume 44, Issue 2, pp 223–232 | Cite as

On the convergence of difference schemes for the approximation of solutionsu ∈ W 2 m (m>0.5) of elliptic equations with mixed derivatives

  • Rajčo D. Lazarov
  • Vladimir L. Makarov
  • Wilfried Weinelt


The paper deals with such estimates of the rate of convergence of difference methods, which are compatible with the smoothness of the exact solutionu ∈ W 2 m (Ω),m>0.5, of elliptic equations with mixed derivatives: The error in the norm of the discrete Sobolev spaceW 2 s (ω), ω denoting the set of grid points, is shown to be of the orderO(|h|m−s), 0≦s<m.

Subject Classifications

AMS: 65N10 CR: 5.17 


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  1. 1.
    Lazarov, R.D., Makarov, V.L., Samarski, A.A.: Application of exact difference schemes for construction and investigation of difference schemes for generalized solutions. Math. Sbornik117 (4), 469–480 (1982) (Russian)Google Scholar
  2. 2.
    Makarov, V.L., Samarski, A.A.: Application of exact difference schemes to error estimates for the method of lines. Žurn. Vyč. Mat. i Mat. Fis.20, (2), 371–387 (Russian)Google Scholar
  3. 3.
    Weinelt, W.: Untersuchungen zur Konvergenzgeschwindigkeit bei Differenzenverfahren. Wiss. Z. Techn. Hochsch. Karl-Marx-Stadt20 (6), 763–769 (1978)Google Scholar
  4. 4.
    Lazarov, R.D.: On the convergence of difference schemes for certain problems of Mathematical Physics with axial symmetry which have generalized solutions. Dokl. Acad. Nauk SSSR258 (6), 1301–1304 (1981) (Russian)Google Scholar
  5. 5.
    Makarov, V.L., Sadullajev, S.S.: On the problem of convergence of difference schemes for the Helmholtz equation in a rectangle. Sbornik “Voprosy Vyč. i Prikl. Mat.”,58, 30–38 (1979) Taschkent 1979 (Russian)Google Scholar
  6. 6.
    Streit, U., Weinelt, W.: Untersuchungen zur Konvergenzordnung von Differenzenmethoden für die Approximation schwacher Lösungen. Anwendung auf eine Stefan-Aufgabe. Wiss. Inform. d. Sekt. Math. Techn. Hochsch. Karl-Marx-Stadt31, 1–54 (1982)Google Scholar
  7. 7.
    Weinelt, W.: Untersuchungen zum Differenzenverfahren für lineare elliptische Variationsungleichungen. Diss. B, Karl-Marx-Stadt 1982Google Scholar
  8. 8.
    Nečas, J.: Sur la coercivite' des formes semilineaires elliptiques. Revue Roumaine Math. Pures Appl.9 (1), 47–69 (1964)Google Scholar
  9. 9.
    Samarski, A.A.: An introduction to the theory of difference schemes. “Nauka”, Moscow 1971 (Russian)Google Scholar
  10. 10.
    Bramble, J.H., Hilbert, S.R.: Bounds for a class of linear functionals with application to Hermite interpolation. Numer. Math.16 (4), 362–369 (1971)Google Scholar
  11. 11.
    Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comput.34 (150), 441–463 (1980)Google Scholar
  12. 12.
    Samarski, A.A., Andrejev, V.B.: Difference methods for elliptic equations. “Nauka”, Moscow 1976 (Russian)Google Scholar
  13. 13.
    Djakonov, E.G.: Difference methods for solving boundary value problems. Moscov Gos. Univ., Moscow 1971 (Russian)Google Scholar
  14. 14.
    Mokin, Ju.I.: Discrete versions of imbedding theorems for classes of the typeW. Žurn. Vyč. Mat. i Mat. Fis.11 (6), 1361–1373 (1971) (Russian)Google Scholar
  15. 15.
    Marčuk, G.I., Agoshkov, V.I.: An introduction to projection-difference methods. “Nauka”, Moscow 1981 (Russian)Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Rajčo D. Lazarov
    • 1
  • Vladimir L. Makarov
    • 2
  • Wilfried Weinelt
    • 3
  1. 1.Institute of MathematicsBulgarian Academy of SciencesSofiaBulgaria
  2. 2.Kiev State UniversityKievUSSR
  3. 3.Sektion Mathematik der Technischen Hochschule Karl-Marx-StadtKarl-Marx-StadtGerman Democratic Republic

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