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\title{An $hp$-adaptive FEM-FCT Algorithm for Convection-Dominated Transport Problems }
\author{} \tocauthor{M. Bittl} \institute{}
\maketitle
\begin{center}
{\large \underline{Melanie Bittl}}\\
University of Erlangen-Nuremberg\\
{\tt melanie.bittl@am.uni-erlangen.de}
\\ \vspace{4mm}{\large Dmitri Kuzmin}\\
University of Erlangen-Nuremberg\\
{\tt kuzmin@am.uni-erlangen.de}
\end{center}
\section*{Abstract}
This talk is concerned with the design of flux limiters for $hp$-adaptive
finite element discretizations of a scalar transport equation. The proposed
approach is based on a continuous Galerkin approximation with unconstrained
high-order elements in smooth regions and constrained $P_1/Q_1$ elements in
the neighborhood of steep fronts. The local mesh size $h$ and polynomial
degree $p$ are chosen using two different error indicators. The first one
is a gradient-based error indicator \cite{zz1987gg} and the second one is a
hierarchical smoothness indicator based on discontinuous higher-order
reconstructions \cite{Kuzmin_Friedhelmgg}. The discrete maximum principle for
linear/bilinear finite elements is enforced using a linearized flux-corrected
transport (FCT) algorithm \cite{Kuzmingg}. The same limiting strategy is employed
when it comes to constraining the $L^2$ projection of data from one
finite-dimensional space into another \cite{Kuzmin_2gg}. The new algorithm is
implemented in the open-source software package HERMES. The use of hierarchical
data structures that support arbitrary level hanging nodes makes the extension
of FCT to $hp$-FEM relatively straightforward. The accuracy of the proposed
method is illustrated by a numerical study for a two-dimensional benchmark
problem with a known exact solution \cite{Kuzmingg,Johngg}.
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{zz1987gg}
{\sc O.C.Zienkiewicz and J.Z.Zhu}. {A simple error estimator and adaptive procedure for practical engineering analysis}. Int. J. Numer. Methods Engrg. 24:2 (1987) 337-357.
\bibitem{Kuzmin_Friedhelmgg}
{\sc D. Kuzmin and F. Schieweck}. {A parameter-free smoothness indicator for high-resolution finite element schemes}. Submitted to the CEJM Topical Issue "Numerical Methods
for Large Scale Scientific Computing" in February 2012.
\bibitem{Kuzmingg}
{\sc D. Kuzmin}. {Explicit and implicit FEM-FCT algorithms with flux linearization}. J. Comput. Phys. 228 (2009) 2517-2534.
\bibitem{Kuzmin_2gg}
{\sc D. Kuzmin, M. Moeller, J.N. Shadid and M. Shashkov}. {Failsafe flux limiting and constrained data projections for equations of gas dynamics}. J. Comput. Phys. 229 (2010) 8766-8779.
\bibitem{Johngg}
{\sc V. John and E. Schmeyer}. {On finite element methods for 3D time-dependent convection-diffusion-reaction equations with small diffusion}. Comput. Meth. Appl. Mech. Engrg. 198 (2008) 475-494.
\end{thebibliography}