On a dimensional reduction method. II. Some approximation-theoretic results

Vogelius, M.; Babuška, I.

doi:10.1090/S0025-5718-1981-0616359-2

On a dimensional reduction method. II. Some approximation-theoretic results

Authors: M. Vogelius and I. Babuška
Journal: Math. Comp. 37 (1981), 47-68
MSC: Primary 65N99; Secondary 65J10
DOI: https://doi.org/10.1090/S0025-5718-1981-0616359-2
MathSciNet review: 616359
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is the second in a series of three that analyze a method of dimensional reduction. It contains some results for approximation of functions on the interval $[ - 1,1]$ with elements from the null-space of ${P^N}$, $N \geqslant 1$, where P is a second-order ordinary differential operator. A special case of this is approximation by polynomials. The one-dimensional results are used as a tool to prove similar versions in several dimensions. These multi-dimensional results are directly related to the approximate method of dimensional reduction that was introduced in [13], and they lead to statements about the convergence properties of this approach. The third paper, which analyzes the adaptive aspects of the method, is forthcoming.

Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/71), 322–333. MR 288971, DOI https://doi.org/10.1007/BF02165003
I. Babuška, B. A. Szabo, and I. N. Katz, The $p$-version of the finite element method, SIAM J. Numer. Anal. 18 (1981), no. 3, 515–545. MR 615529, DOI https://doi.org/10.1137/0718033
M. S. Baouendi and C. Goulaouic, Régularité et théorie spectrale pour une classe d’opérateurs elliptiques dégénérés, Arch. Rational Mech. Anal. 34 (1969), 361–379 (French). MR 249844, DOI https://doi.org/10.1007/BF00281438

J. Bergh & J. Löfström, Interpolation Spaces, Springer-Verlag, Berlin and New York, 1976.

P. Destuynder, Sur une Justification des Modeles de Plaques et de Coques par les Méthodes Asymptotiques, Doctoral Thesis, Paris VI, January 1980.

V. K. Dzyadyk, Vvedenie v teoriyu ravnomernogo priblizheniya funktsiĭ polinomami, Izdat. “Nauka”, Moscow, 1977 (Russian). MR 0612836

D. G. Gordeziani, "Accuracy of a variant in the theory of thin shells," Soviet Phys. Dokl., v. 19, 1974, pp. 385-386.

J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR 0350177
G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. MR 0213785

S. M. Nikol’skiĭ, Approximation of Functions of Several Variables and Imbedding Theorems, Springer-Verlag, Berlin and New York, 1975.

F. G. Tricomi, Differential Equations, Blackie & Son, London, 1961.

M. Vogelius, A Dimensional Reduction Approach to the Solution of Partial Differential Equations, Ph. D. Thesis, University of Maryland, December 1979.

M. Vogelius and I. Babuška, On a dimensional reduction method. I. The optimal selection of basis functions, Math. Comp. 37 (1981), no. 155, 31–46. MR 616358, DOI https://doi.org/10.1090/S0025-5718-1981-0616358-0