Finite element error estimates for nonlinear convective problems

• ID: 2735, RIV: 10331078
• ISSN: 1570-2820, ISBN: not specified
• source: Journal of Numerical Mathematics
• keywords: nonlinear convection equation; finite element method; a priori error estimates; continuous mathematical induction; continuation
• authors: Václav Kučera
• authors from KNM: Kučera Václav

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.