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Mathematics of Computation

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



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
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.

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  • 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, Differentsialnye sootnosheniya s chastnymi proizvodnymi, “Mir”, Moscow, 1990 (Russian). Translated from the English and with a preface and an appendix by N. M. Mishachëv. MR 1065393
  • 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. Tihomirov, 𝜖-entropy and 𝜖-capacity of sets in functional space, Amer. Math. Soc. Transl. (2) 17 (1961), 277–364. MR 0124720
  • 6. B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, No. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR 0212575
  • 7. Koichi Niijima, A posteriori error bounds for piecewise linear approximate solutions of elliptic equations of monotone type, Math. Comp. 58 (1992), no. 198, 549–560. MR 1122073, 10.1090/S0025-5718-1992-1122073-7
  • 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, Internat. J. Numer. Methods Engrg. 32 (1991), no. 3, 471–499. MR 1121798, 10.1002/nme.1620320303
  • 9. I. S. Krasil′shchik, V. V. Lychagin, and A. M. Vinogradov, Geometry of jet spaces and nonlinear partial differential equations, Advanced Studies in Contemporary Mathematics, vol. 1, Gordon and Breach Science Publishers, New York, 1986. Translated from the Russian by A. B. Sosinskiĭ. MR 861121
  • 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. Lothar Collatz, Numerische Behandlung von Differentialgleichungen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Band LX, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1951 (German). MR 0043563

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Additional Information

Z. Wiener
Affiliation: Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel

Y. Yomdin
Affiliation: Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel

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