Finite element error estimates for nonlinear convective problems

• ID: 2584, RIV: 10129881
• ISSN: not specified, ISBN: 978-80-227-3742-5
• source: ALGORITMY 2012, 19th Conference on Scientific Computing
• 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 nonstationary nonlinear convective problems. Using special estimates of the convective terms, we prove apriori error estimates for a semidiscrete and implicit scheme. For the semidiscrete scheme we need to apply so-called continuous mathematical induction and a nonlinear Gronwall lemma. For the implicit scheme, we prove that there does not exist a Gronwall-type lemma capable of proving the desired estimates using standard arguments. To overcome this obstacle, we use a suitable continuation of the discrete implicit solution and again use continuous mathematical induction to prove the error estimates. The technique presented can be extended to locally Lipschitz-continuous convective nonlinearities.