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Mathematics of Computation

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From formal numerical solutions
of elliptic PDE's to the true ones


Authors: Z. Wiener and Y. Yomdin
Journal: Math. Comp. 69 (2000), 197-235
MSC (1991): Primary 65N06, 65N15, 35J05
DOI: https://doi.org/10.1090/S0025-5718-99-01147-3
Published electronically: August 19, 1999
MathSciNet review: 1654018
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Abstract | References | Similar Articles | Additional Information

Abstract: We propose a discretization scheme for a numerical solution of elliptic PDE's, based on local representation of functions, by their Taylor polynomials (jets). This scheme utilizes jet calculus to provide a very high order of accuracy for a relatively small number of unknowns involved.


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

  • 1. E. Bichuch, and Y. Yomdin, Numerical solution of parabolic equation by the method of high order discretization, The Weizmann Institute of Science, preprint (1994).
  • 2. M. Gromov, Partial Differential Relations, A Series of Modern Surveys in Mathematics 3 Folge, Band 9, Springer-Verlag, 1986. MR 91e:53047
  • 3. E. Kochavi, R. Segev, and Y. Yomdin, Numerical Solution of Field Problems by Nonconforming Taylor Discretization., Applied Mathematical Modeling 15 (1991), 152-157.
  • 4. E. Kochavi, R. Segev, and Y. Yomdin, Modified algorithms for nonconforming Taylor discretization, Computers and Structures 49 (6) (1993), 969-979.
  • 5. A. N. Kolmogorov, and V. M. Tikomoirov, $\epsilon $-entropy and $\epsilon $-capacity of sets in functional spaces, AMS Transl. 17 (2) (1961), 277-364. MR 23:A2031
  • 6. B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research, Bombay and Oxford University press, 1966. MR 35:3446
  • 7. K. Niijima, A posteriory error bounds for piecewise linear approximate solutions of elliptic equations of monotone type, Math. Comp. 58 (1992), 549-560. MR 92g:65112
  • 8. P. J. M. Sonnemans, L. P. H. De Goey, and J. K. Nieuwenhuizen, Optimal Use of a Numerical Method for Solving Differential Equations Based on Taylor Series Expansions., Intern. J. for Numerical Methods in Engeneering 32 (1991), 471-499. MR 92d:65123
  • 9. A. M. Vinogradov, I. S. Krasilshik and V. V. Lychagin, Geometry of jet-spaces and nonlinear differential equations, Moscow, 1977 (Russian).MR 88m:58211
  • 10. H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63-89.
  • 11. Y. Yomdin, Y. Elihay, Flexible high order discretization, preprint.
  • 12. L. Collatz, Numerische Behandlung von Differentialgleichungen, Springer, 1951.MR 13:285f

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Additional Information

Z. Wiener
Affiliation: Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
Email: mswiener@pluto.mscc.huji.ac.il

Y. Yomdin
Affiliation: Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
Email: yomdin@wisdom.weizmann.ac.il

DOI: https://doi.org/10.1090/S0025-5718-99-01147-3
Received by editor(s): October 14, 1994
Received by editor(s) in revised form: May 23, 1997
Published electronically: August 19, 1999
Article copyright: © Copyright 1999 American Mathematical Society

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