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Journal of Numerical Mathematics

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

Managing Editor: Olshanskii, Maxim

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Volume 24, Issue 3


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
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Published Online: 2016-10-05 | DOI: https://doi.org/10.1515/jnma-2015-0030


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


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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.

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