The stability of Ritz-Volterra projection and error estimates for finite element methods for a class of integro-differential equations of parabolic type

Yan Ping Lin; Tie Zhu Zhang

Applications of Mathematics (1991)

  • Volume: 36, Issue: 2, page 123-133
  • ISSN: 0862-7940

Abstract

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In this paper we first study the stability of Ritz-Volterra projection (see below) and its maximum norm estimates, and then we use these results to derive some L ∞ error estimates for finite element methods for parabolic integro-differential equations.

How to cite

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Lin, Yan Ping, and Zhang, Tie Zhu. "The stability of Ritz-Volterra projection and error estimates for finite element methods for a class of integro-differential equations of parabolic type." Applications of Mathematics 36.2 (1991): 123-133. <http://eudml.org/doc/15664>.

@article{Lin1991,
abstract = {In this paper we first study the stability of Ritz-Volterra projection (see below) and its maximum norm estimates, and then we use these results to derive some $L^\infty $ error estimates for finite element methods for parabolic integro-differential equations.},
author = {Lin, Yan Ping, Zhang, Tie Zhu},
journal = {Applications of Mathematics},
keywords = {Ritz-Volterra projection; stability; finite element; error estimates; initial- boundary-value problem; parabolic Volterra integro-differential equation; stability; finite element methods; Ritz-Volterra projection; initial- boundary-value problem; parabolic Volterra integro-differential equation; error estimates},
language = {eng},
number = {2},
pages = {123-133},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The stability of Ritz-Volterra projection and error estimates for finite element methods for a class of integro-differential equations of parabolic type},
url = {http://eudml.org/doc/15664},
volume = {36},
year = {1991},
}

TY - JOUR
AU - Lin, Yan Ping
AU - Zhang, Tie Zhu
TI - The stability of Ritz-Volterra projection and error estimates for finite element methods for a class of integro-differential equations of parabolic type
JO - Applications of Mathematics
PY - 1991
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 36
IS - 2
SP - 123
EP - 133
AB - In this paper we first study the stability of Ritz-Volterra projection (see below) and its maximum norm estimates, and then we use these results to derive some $L^\infty $ error estimates for finite element methods for parabolic integro-differential equations.
LA - eng
KW - Ritz-Volterra projection; stability; finite element; error estimates; initial- boundary-value problem; parabolic Volterra integro-differential equation; stability; finite element methods; Ritz-Volterra projection; initial- boundary-value problem; parabolic Volterra integro-differential equation; error estimates
UR - http://eudml.org/doc/15664
ER -

References

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  2. J. R. Cannon Y. Lin, 10.1137/0727036, SJAM. J. Numer. Anal., 27 (1990) 595-607. (1990) MR1041253DOI10.1137/0727036
  3. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, 1978. (1978) Zbl0383.65058MR0520174
  4. E. Green-Yanik G. Fairweather, Finite element methods for parabolic and hyperbolic partial integro-differential equations, to appear in Nonlinear Analysis. MR0954953
  5. M. N. Le Roux V. Thomee, 10.1137/0726075, SIAM J. Numer. Anal., 26 (1989) 1291-1309. (1989) MR1025089DOI10.1137/0726075
  6. Y. Lin V. Thomee L. Wahlbin, A Ritz-Volterra projection onto finite element spaces and application to integro and related equations, to appear in SIAM J. Numer. Anal. MR1111453
  7. Qun Lin, Tao Lu, Shu-min Shen, Maximum norm estimate, extrapolation and optimal points of stresses for the finite element methods on the strongly regular triangulalion, J. Соmр. Math., Vol. 1, No. 4 (1983) 376-383. (1983) MR0726394
  8. Qun Lin, Qi-ding Zhou, Superconvergence Theory of Finite Element Methods, Book to appear. 
  9. J. A. Nitsche, L ∞ -convergence of finite element Galerkin approximations for parabolic problems, R.A.I.R.O., Vol. 13, No. 1, (1979) 31-51. (1979) Zbl0401.65069MR0527037
  10. R. Rannacher R. Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Соmр. 38 (1982) 437-445. (1982) MR0645661
  11. A. H. Schatz V. Thomée L. Wahlbin, Maximum norm stability and error estimates in parabolic finite element equations, Comm. Pur. Appl. Math., XXXIII, (1980) 265-304. (1980) MR0562737
  12. R. Scott, Optimal L ∞ estimates for the finite element on irregular meshes, Math. Соmр., 30 (1976) 681-697. (1976) Zbl0349.65060MR0436617
  13. V. Thomee N. Y. Zhang, Error estimates for semi-discrete finite element methods for parabolic integro-differential equations, Math. Соmр., 53 (1989) 121-139. (1989) MR0969493
  14. M. F. Wheeler, 10.1137/0710062, SIAM J. Numer. Anal. 19 (1973) 723-759. (1973) MR0351124DOI10.1137/0710062

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