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On a dimensional reduction method. I. The optimal selection of basis functions


Authors: M. Vogelius and I. Babuška
Journal: Math. Comp. 37 (1981), 31-46
MSC: Primary 65N99; Secondary 65J10
DOI: https://doi.org/10.1090/S0025-5718-1981-0616358-0
MathSciNet review: 616358
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper is the first in a series of three, which analyze an adaptive approximate approach for solving $ (n + 1)$-dimensional boundary value problems by replacing them with systems of equations in n-dimensional space. In this approach the unknown functions of $ (n + 1)$ variables are projected onto finite linear combinations of functions of just n variables.

This paper shows how the coefficients of these linear combinations can be chosen optimally.


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

  • [1] I. Babuška & W. C. Rheinboldt, "Mathematical problems of computational decisions for the finite element method," in Mathematical Aspects of Finite Element Methods (I. Galligani and E. Magenes, Eds.), Lecture Notes in Math., vol. 606, Springer-Verlag, Berlin and New York, 1977, pp. 1-26. MR 0483538 (58:3533)
  • [2] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Groningen, 1976. MR 0390843 (52:11666)
  • [3] J. Bergh & J. LöfstrÖm, Interpolation Spaces, Springer-Verlag, Berlin and New York, 1976.
  • [4] V. E. Chepiga, "On constructing a theory of multilayer anisotropic shells with prescribed arbitrary accuracy of order $ {h^N}$," Mech. Solids, v. 12, 1977, pp. 113-120.
  • [5] P. G. Ciarlet & P. Destuynder, "A justification of the two-dimensional linear plate model," J. Mécanique, v. 18, 1979, pp. 315-344. MR 533827 (80e:73046)
  • [6] N. Dunford & J. Schwartz, Linear Operators, Part I, Interscience, New York, 1958.
  • [7] K. O. Friedrichs & R. F. Dressler, "A boundary layer theory for elastic plates," Comm. Pure. Appl. Math., v. 14, 1961, pp. 1-33. MR 0122117 (22:12844)
  • [8] A. L. Gol'denweizer, "Derivation of an approximate theory of bending of a plate by the method of asymptotic integration of the equations of the theory of elasticity," J. Appl. Math. Mech., v. 26, 1962, pp. 1000-1025. MR 0170523 (30:761)
  • [9] L. V. Kantorovich & V. I. Krylov, Approximate Methods of Higher Analysis, Noordhoff, Groningen, 1958. MR 0106537 (21:5268)
  • [10] J. L. Lions, Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal, Lecture Notes in Math., vol. 323, Springer-Verlag, Berlin and New York, 1973. MR 0600331 (58:29078)
  • [11] P. M. Naghdi, Handbuch der Physik, Band V $ {\text{a}}/2$, Springer-Verlag, Berlin and New York, 1972, pp. 425-640.
  • [12] V. V. Poniatovskiĭ, "Theory for plates of medium thickness," J. Appl. Math. Mech., v. 26, 1962, pp. 335-341. MR 0151022 (27:1008)
  • [13] F. Riesz & B. Sz.-Nagy, Functional Analysis, Ungar, New York, 1955. MR 0071727 (17:175i)
  • [14] M. Vogelius, A Dimensional Reduction Approach to the Solution of Partial Differential Equations, Ph. D. Thesis, University of Maryland, Dec. 1979.
  • [15] M. Vogelius & I. Babuška, "On a dimensional reduction method. II. Some approximation-theoretic results," Math. Comp., v. 37, 1981, pp. 47-68. MR 616359 (83c:65259b)
  • [16] M. Vogelius & I. Babuška, "On a dimensional reduction method. III. A posteriori error estimation and an adaptive approach," Math. Comp. (To appear.) MR 628701 (83c:65259c)
  • [17] I. E. Zino & E. A. Tropp, Asymptotic Methods in the Problems of Heat Transfer and Thermoelasticity, Univ. of Leningrad Publ., Leningrad, 1978. (Russian)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1981-0616358-0
Article copyright: © Copyright 1981 American Mathematical Society

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