Finite element error estimates for nonlinear convective problems

Abstract

This paper is concerned with the analysis of the nite 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.