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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On the convergence rates of variational methods. I. Asymptotically diagonal systems
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by L. M. Delves and K. O. Mead PDF
Math. Comp. 25 (1971), 699-716 Request permission

Abstract:

We consider the problem of estimating the convergence rate of a variational solution to an inhomogeneous equation. This problem is not soluble in general without imposing conditions on both the class of expansion functions and the class of problems considered; we introduce the concept of “asymptotically diagonal systems,” which is particularly appropriate for classical variational expansions as applied to elliptic partial differential equations. For such systems, we obtain a number of a priori estimates of the asymptotic convergence rate which are easy to compute, and which are likely to be realistic in practice. In the simplest cases these estimates reduce the problem of variational convergence to the simpler problem of Fourier series convergence, which is considered in a companion paper. We also produce estimates for the convergence rate of the individual expansion coefficients $a_i^{(n)}$, thus categorising the convergence completely.
References
  • L. V. Kantorovič and V. I. Krylov, Približennye metody vysšego analiza, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (Russian). 3d ed.]. MR 0042210
  • S. G. Mikhlin, Variational methods in mathematical physics, A Pergamon Press Book, The Macmillan Company, New York, 1964. Translated by T. Boddington; editorial introduction by L. I. G. Chambers. MR 0172493
  • C. Schwartz, Estimating Convergence Rates of Variational Calculations, Methods in Computational Physics, vol. 2, Academic Press, New York, 1963.
  • J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Math. Comp. 25 (1971), 699-716
  • MSC: Primary 65N30
  • DOI: https://doi.org/10.1090/S0025-5718-1971-0311131-9
  • MathSciNet review: 0311131