Finite element error estimates for nonlinear convective problems

Václav Kučera

doi:10.1515/jnma-2015-0030

Finite element error estimates for nonlinear convective problems

2016-01-01

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Václav Kučera *

DOI：10.1515/jnma-2015-0030

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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.

Keyword:

nonlinear convection equation

finite element method

a priori error estimates

continuous mathematical induction

continuation

Finite element error estimates for nonlinear convective problems

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