Abstract
Assume that for the approximate solution of an elliptic differential equation in a bounded domain Ω, under a natural boundary condition, one applies the Galerkin method with polynomial coordinate functions. One gives sufficient conditions, imposed on the exact solutionu *, which ensure the convergence of the derivatives of order k of the approximate solutions, uniformly or in the mean in Ω or in any interior subdomain. For example, ifu *∈Wk 2, then the derivatives of order k converge in L2(Ω′), where Ω′ is an interior subdomain of Ω. Somewhat weaker statements are obtained in the case of the Dirchlet problem.
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 70, pp. 11–18, 1977.
The author expresses his gratitude to Yu. K. Dem'yanovich for drawing his attention to [10].
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Daugavet, I.K. Convergence of the highest derivatives in projection methods. J Math Sci 23, 1878–1884 (1983). https://doi.org/10.1007/BF01093272
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DOI: https://doi.org/10.1007/BF01093272