Summary
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 m2 (Ω),m>0.5, of elliptic equations with mixed derivatives: The error in the norm of the discrete Sobolev spaceW s2 (ω), ω denoting the set of grid points, is shown to be of the orderO(|h| m−s), 0≦s<m.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
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)
Weinelt, W.: Untersuchungen zur Konvergenzgeschwindigkeit bei Differenzenverfahren. Wiss. Z. Techn. Hochsch. Karl-Marx-Stadt20 (6), 763–769 (1978)
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)
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)
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)
Weinelt, W.: Untersuchungen zum Differenzenverfahren für lineare elliptische Variationsungleichungen. Diss. B, Karl-Marx-Stadt 1982
Nečas, J.: Sur la coercivite' des formes semilineaires elliptiques. Revue Roumaine Math. Pures Appl.9 (1), 47–69 (1964)
Samarski, A.A.: An introduction to the theory of difference schemes. “Nauka”, Moscow 1971 (Russian)
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)
Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comput.34 (150), 441–463 (1980)
Samarski, A.A., Andrejev, V.B.: Difference methods for elliptic equations. “Nauka”, Moscow 1976 (Russian)
Djakonov, E.G.: Difference methods for solving boundary value problems. Moscov Gos. Univ., Moscow 1971 (Russian)
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)
Marčuk, G.I., Agoshkov, V.I.: An introduction to projection-difference methods. “Nauka”, Moscow 1981 (Russian)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lazarov, R.D., Makarov, V.L. & Weinelt, W. On the convergence of difference schemes for the approximation of solutionsu ∈ W m2 (m>0.5) of elliptic equations with mixed derivatives. Numer. Math. 44, 223–232 (1984). https://doi.org/10.1007/BF01410107
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01410107