Some convergence problems for weak norms
HTML articles powered by AMS MathViewer
- by
I. K. Daugavet
Translated by: A. Plotkin - St. Petersburg Math. J. 15 (2004), 575-585
- DOI: https://doi.org/10.1090/S1061-0022-04-00823-4
- Published electronically: July 6, 2004
- PDF | Request permission
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 \[ U^{(n)}=\{ u\in U \mid \chi (u)=0, \chi \in U_n^*\}. \] If the sequence $\{U_n^*\}$ is asymptotically dense in $U^*$, then $\|I_n\|\to 0$, 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 $\|u-P_nu\|_V/\|u-P_nu\|_U\to 0$. The norm $\|I_n\|$ 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
- 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 295599
- Lars B. Wahlbin, Superconvergence in Galerkin finite element methods, Lecture Notes in Mathematics, vol. 1605, Springer-Verlag, Berlin, 1995. MR 1439050, DOI 10.1007/BFb0096835
- N. P. Korneĭchuk, Tochnye konstanty v teorii priblizheniya, “Nauka”, Moscow, 1987 (Russian). MR 926687
- 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)
- S. G. Mikhlin, Nekotorye voprosy teorii pogreshnosteĭ , Leningrad. Univ., Leningrad, 1988 (Russian). MR 964471
- 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 455485
- I. K. Daugavet, Vvedenie v teoriyu priblizheniya funktsiĭ, Izdat. Leningrad. Univ., Leningrad, 1977 (Russian). MR 0470560
- —, Approximate solution of linear functional equations, Leningrad. Univ., Leningrad, 1985. (Russian)
- 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
- I. K. Daugavet, On the method of moments for ordinary differential equations, Sibirsk. Mat. Ž. 6 (1965), 70–85 (Russian). MR 0173807
Bibliographic Information
- I. K. Daugavet
- Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 28, St. Petersburg 198504, Russia
- Email: daugavet@vd3999.spb.edu
- Received by editor(s): December 18, 2002
- Published electronically: July 6, 2004
- © Copyright 2004 American Mathematical Society
- Journal: St. Petersburg Math. J. 15 (2004), 575-585
- MSC (2000): Primary 46N40
- DOI: https://doi.org/10.1090/S1061-0022-04-00823-4
- MathSciNet review: 2068983