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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Some convergence problems for weak norms

Author: I. K. Daugavet
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 15 (2003), nomer 4.
Journal: St. Petersburg Math. J. 15 (2004), 575-585
MSC (2000): Primary 46N40
Published electronically: July 6, 2004
MathSciNet review: 2068983
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $U$ be a normed space compactly embedded in a space $V$, let $\{U_n^*\}$ be a sequence of finite-dimensional subspaces of the dual space $U^*$, and let

\begin{displaymath}U^{(n)}=\{ u\in U \mid \chi(u)=0, \chi\in U_n^*\}. \end{displaymath}

If the sequence $\{U_n^*\}$ is asymptotically dense in $U^*$, then $\Vert I_n\Vert\to 0$, where $I_n$ denotes the operator that embeds $U^{(n)}$ in $V$. In particular, if $\{P_n\}$ is a sequence of finite-dimensional projections in $U$, and the sequence $\{{\mathcal R}(P_n^*)\}$ is asymptotically dense in $U^*$, then $\Vert u-P_nu\Vert _V/\Vert u-P_nu\Vert _U\to0$. The norm $\Vert I_n\Vert$ is estimated in terms of the best approximation of the elements of the unit ball in $V^*$ (this ball is compact in $U^*$) by elements of $U_n^*$. Usually, for projection methods of solving functional equations, the metric in which the convergence should be studied is dictated by the general convergence theorems (we mean, e.g., the energy metric for the Ritz method). The above arguments make it possible to establish a faster convergence of projection methods in weaker metrics. Some results of this type are obtained in the paper for the Ritz and Galerkin methods and for the method of moments.

References [Enhancements On Off] (What's this?)

  • 1. L. A. Oganesjan and L. A. Ruhovec, An investigation of the rate of convergence of variation-difference schemes for second order elliptic equations in a two-dimensional region with smooth boundary, Ž. Vyčisl. Mat. i Mat. Fiz. 9 (1969), 1102–1120 (Russian). MR 0295599
  • 2. Lars B. Wahlbin, Superconvergence in Galerkin finite element methods, Lecture Notes in Mathematics, vol. 1605, Springer-Verlag, Berlin, 1995. MR 1439050
  • 3. N. P. Korneĭchuk, \cyr Tochnye konstanty v teorii priblizheniya, “Nauka”, Moscow, 1987 (Russian). MR 926687
  • 4. I. Yu. Kharrik, Approximation of functions which vanish on the boundary of a region, together with their partial derivatives, by functions of a special type, Sibirsk. Mat. Zh. 4 (1963), no. 2, 408-425. (Russian) MR 0000000 (27:517)
  • 5. S. G. Mikhlin, \cyr Nekotorye voprosy teorii pogreshnosteĭ, Leningrad. Univ., Leningrad, 1988 (Russian). MR 964471
  • 6. T. O. Šapošnikova, A priori error estimates of variational methods in Banach spaces, Ž. Vyčisl. Mat. i Mat. Fiz. 17 (1977), no. 5, 1144–1152, 1332 (Russian). MR 0455485
  • 7. I. K. Daugavet, \cyr Vvedenie v teoriyu priblizheniya funktsiĭ., Izdat. Leningrad. Univ., Leningrad, 1977 (Russian). MR 0470560
  • 8. -, Approximate solution of linear functional equations, Leningrad. Univ., Leningrad, 1985. (Russian)
  • 9. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
  • 10. I. K. Daugavet, On the method of moments for ordinary differential equations, Sibirsk. Mat. Ž. 6 (1965), 70–85 (Russian). MR 0173807

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Additional Information

I. K. Daugavet
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 28, St. Petersburg 198504, Russia

Keywords: Superconvergence, projection, projection methods, Ritz and Galerkin methods, method of moments
Received by editor(s): December 18, 2002
Published electronically: July 6, 2004
Article copyright: © Copyright 2004 American Mathematical Society