Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Numerical Mathematics

Editor-in-Chief: Hoppe, Ronald H. W. / Kuznetsov, Yuri

Managing Editor: Olshanskii, Maxim

Editorial Board: Benzi, Michele / Brenner, Susanne C. / Carstensen, Carsten / Dryja, M. / Feistauer, Miloslav / Glowinski, R. / Lazarov, Raytcho / Nataf, Frédéric / Neittaanmaki, P. / Bonito, Andrea / Quarteroni, Alfio / Guzman, Johnny / Rannacher, Rolf / Repin, Sergey I. / Shi, Zhong-ci / Tyrtyshnikov, Eugene E. / Zou, Jun / Simoncini, Valeria / Reusken, Arnold

4 Issues per year


IMPACT FACTOR 2017: 0.951
5-year IMPACT FACTOR: 3.128

CiteScore 2017: 0.96

SCImago Journal Rank (SJR) 2017: 0.494
Source Normalized Impact per Paper (SNIP) 2017: 0.772

Mathematical Citation Quotient (MCQ) 2017: 1.68

Online
ISSN
1569-3953
See all formats and pricing
More options …
Volume 24, Issue 3

Issues

Finite element error estimates for nonlinear convective problems

Václav Kučera
  • Corresponding author
  • Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, Praha 8, 186 75, Czech Republic
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-10-05 | DOI: https://doi.org/10.1515/jnma-2015-0030

Abstract

This paper is concerned with the analysis of the finite element method applied to a nonstationary nonlinear convective problem. Using special estimates of the convective terms, we prove a priori error estimates for an explicit, semidiscrete and implicit scheme. While the explicit case is rather straightforward via mathematical induction, for the semidiscrete scheme we need to apply so-called continuous mathematical induction and a nonlinear Gronwall lemma. For the implicit scheme, we use a suitable continuation of the discrete implicit solution and again use continuous mathematical induction to prove the error estimates. Finally, we extend the presented analysis from globally Lipschitz-continuous convective nonlinearities to the locally Lipschitz-continuous case.

Keywords: nonlinear convection equation; finite element method; a priori error estimates; continuous mathematicalinduction; continuation

MSC 2010: 65M15; 65M60; 65M12

References

  • [1]

    R. Bank, J. B. Burger, W. Fichtner, and R. Smith, Some up-winding techniques for finite element approximations of convection diffusion equation, Numer. Math. 58 (1990), 185–202.Google Scholar

  • [2]

    Y. R. Chao, A note on ‘Continuous mathematical induction’, Bull. Amer. Math. Soc. 26 (1919), 17–18.Google Scholar

  • [3]

    P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1979.Google Scholar

  • [4]

    P. L. Clark, Real induction, http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.187.3514.

  • [5]

    M. Dobrowolski and H.-G. Roos, A priori estimates for the solution of convection–diffusion problems and interpolation on Shishkin meshes, Z. Anal. Anwend., 16 (1997), 1001–1012.Google Scholar

  • [6]

    V. Dolejší, M. Feistauer, and J. Hozman, Analysis of semi-implicit DGFEM for nonlinear convection–diffusion problems on nonconforming meshes, Comput. Meth. Appl. Mech. Engrg., 197 (2007), 2813–2827.Google Scholar

  • [7]

    M. Feistauer, J. Felcman, and I. Straškraba, Mathematical and Computational Methods for Compressible Flow, Clarendon Press, Oxford, 2003.Google Scholar

  • [8]

    V. Girault and P. A. Raviart, Finite Element Methods for Navier Stokes Equations, Theorems and Algorithms, Springer Verlag, 1986.Google Scholar

  • [9]

    T. J. R. Hughes and A. Brooks, A multi-dimension upwind scheme with no crosswind diffusion, in: AMD 34 (1979), Am. Soc. Mech. Engrg., New York, 19–35.Google Scholar

  • [10]

    J. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, 1987.Google Scholar

  • [11]

    V. Kučera, On diffusion-uniform error estimates for the DG method applied to singularly perturbed problems, IMA J. Numer. Anal. 34 (2014), 820–861.Google Scholar

  • [12]

    A. Kufner, O. John and S. Fučík, Function Spaces, Academia, Prague, 1977.Google Scholar

  • [13]

    X.G. Li, C.K. Chan, and S. Wang, The finite element method with weighted basis functions for singularly perturbed convection–diffusion problems, J. Comput. Phys., 195 (2004), 773–789.Google Scholar

  • [14]

    H.-G. Roos, M. Stynes, and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer-Verlag, Berlin–Heidelberg, 2008.Google Scholar

  • [15]

    E. Zeidler, Nonlinear Functional Analysis and Its Applications II/B: Nonlinear Monotone Operators, Springer, 1986.Google Scholar

  • [16]

    Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws, SIAMJ. Numer. Anal., 42 (2004), 641–666.Google Scholar

About the article

Received: 2015-03-09

Revised: 2015-09-14

Accepted: 2015-10-09

Published Online: 2016-10-05

Published in Print: 2016-10-01


Citation Information: Journal of Numerical Mathematics, Volume 24, Issue 3, Pages 143–165, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI: https://doi.org/10.1515/jnma-2015-0030.

Export Citation

© 2016 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in