# hpfem/esco2012-boa

### Subversion checkout URL

You can clone with
or
.
Fetching contributors…

Cannot retrieve contributors at this time

29 lines (18 sloc) 2.315 kB
 \title{Error Estimates for Nonlinear, Purely Convective Problems in Finite Element Methods} \tocauthor{V. Kucera} \author{} \institute{} \maketitle \begin{center} {\large V\'{a}clav Ku\v{c}era}\\ Charles University in Prague, Faculty of Mathematics and Physics\\ {\tt vaclav.kucera@email.cz} \end{center} \section*{Abstract} This paper is concerned with the analysis of finite element methods (standard conforming FEM and discontinuous Galerkin) applied to a nonstationary nonlinear convective problem with mixed Dirichlet-Neumann boundary conditions. Using the technique introduced in \cite{ZhangShu}, we prove apriori error estimates for higher order finite element discretizations of such an equation, assuming the exact solution is sufficiently regular on some time interval $[0,T]$. In previous work, nonlinear convection was treated mainly in the context of convection-diffusion problems and the use of parabolic techniques resulted in error estimates with an exponential blow up with respect to the diffusion coefficient going to zero, cf. \cite{Moje}. In \cite{ZhangShu}, the authors perform their analysis for various explicit schemes using an argument which relies heavily on mathematical induction. We extend the analysis to the method of lines using so-called continuous mathematical induction and a nonlinear Gronwall-type lemma. For a fully implicit scheme, we prove that there does not exist a Gronwall-type lemma capable of proving the desired estimates using standard arguments. Next, we use a suitable continuation of the discrete implicit solution and again use continuous mathematical induction to prove error estimates under a CFL-like condition. Finally, we extend the analysis from globally Lipschitz continuous convective nonlinearities to the locally Lipschitz continuous case. \bibliographystyle{plain} \begin{thebibliography}{10} \bibitem{ZhangShu} {\sc Q. Zhang and C.-W. Shu}. {Error Estimates to Smooth Solutions of Runge-Kutta Discontinuous Galerkin Methods for Scalar Conservation Laws}. SIAM J. Numer. Anal. 42, 2 (2004) 641-666. \bibitem{Moje} {\sc V. Ku\v{c}era}. {Optimal $L^\infty(L^2)$-error Estimates for the DG Method Applied to Nonlinear Convection-Diffusion Problems with Nonlinear Diffusion}. Numer. Func. Anal. Opt. 31, 3 (2010) 285-312. \end{thebibliography}
Something went wrong with that request. Please try again.