# hpfem/esco2012-boa

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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}