Estimates of deviations from exact solutions of initial-boundary value problem for the heat equation

Sergey Repin

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2002)

  • Volume: 13, Issue: 2, page 121-133
  • ISSN: 1120-6330

Abstract

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The paper is concerned with deriving functionals that give upper bounds of the difference between the exact solution of the initial-boundary value problem for the heat equation and any admissible function from the functional class naturally associated with this problem. These bounds are given by nonegative functionals called deviation majorants, which vanish only if the function and exact solution coincide. The deviation majorants pose a new type of a posteriori estimates that can be used in numerical analysis. Also, the estimates formed by such majorants can be viewed as a certain extension of well known «energy» estimates for solutions of parabolic type problems (see [1]).

How to cite

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Repin, Sergey. "Estimates of deviations from exact solutions of initial-boundary value problem for the heat equation." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 13.2 (2002): 121-133. <http://eudml.org/doc/252364>.

@article{Repin2002,
abstract = {The paper is concerned with deriving functionals that give upper bounds of the difference between the exact solution of the initial-boundary value problem for the heat equation and any admissible function from the functional class naturally associated with this problem. These bounds are given by nonegative functionals called deviation majorants, which vanish only if the function and exact solution coincide. The deviation majorants pose a new type of a posteriori estimates that can be used in numerical analysis. Also, the estimates formed by such majorants can be viewed as a certain extension of well known «energy» estimates for solutions of parabolic type problems (see [1]).},
author = {Repin, Sergey},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Parabolic equations; Deviations from exact solution; A posteriori estimates; parabolic equations; deviations from exact solution; a posteriori estimates},
language = {eng},
month = {6},
number = {2},
pages = {121-133},
publisher = {Accademia Nazionale dei Lincei},
title = {Estimates of deviations from exact solutions of initial-boundary value problem for the heat equation},
url = {http://eudml.org/doc/252364},
volume = {13},
year = {2002},
}

TY - JOUR
AU - Repin, Sergey
TI - Estimates of deviations from exact solutions of initial-boundary value problem for the heat equation
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2002/6//
PB - Accademia Nazionale dei Lincei
VL - 13
IS - 2
SP - 121
EP - 133
AB - The paper is concerned with deriving functionals that give upper bounds of the difference between the exact solution of the initial-boundary value problem for the heat equation and any admissible function from the functional class naturally associated with this problem. These bounds are given by nonegative functionals called deviation majorants, which vanish only if the function and exact solution coincide. The deviation majorants pose a new type of a posteriori estimates that can be used in numerical analysis. Also, the estimates formed by such majorants can be viewed as a certain extension of well known «energy» estimates for solutions of parabolic type problems (see [1]).
LA - eng
KW - Parabolic equations; Deviations from exact solution; A posteriori estimates; parabolic equations; deviations from exact solution; a posteriori estimates
UR - http://eudml.org/doc/252364
ER -

References

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  10. Repin, S., A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comput., v. 69, 230, 2000, 481-500. Zbl0949.65070MR1681096DOI10.1090/S0025-5718-99-01190-4
  11. Repin, S., A unified approach to a posteriori error estimation based on duality error majorants. Mathematics and Computers in Simulation, 50, 1999, 313-329. MR1717590DOI10.1016/S0378-4754(99)00081-6
  12. Repin, S., Estimates of the accuracy of two-dimensional models in elasticity theory. Probl. Mat. Anal., v. 22, 2001, 178-196 (in Russian). Zbl1150.74368
  13. Repin, S., Two-sided estimates of deviations from exact solutions of uniformly elliptic equations. Trudi St.-Petersburg Mathematickal Society, v. 9, 2001, 148-179. Zbl1039.65076MR2018375
  14. Repin, S., A posteriori estimates of the accuracy of variational methods for problems with nonconvex functionals. Algebra i Analiz, 11, 1999, n. 4, 151-182 (in Russian, translated in St.-Petersburg Mathematical Journal, v. 11, n. 4, 2000). Zbl0964.49009MR1713937

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