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\title{An $hp$-adaptive FEM-FCT Algorithm for Convection-Dominated Transport Problems }
\author{} \tocauthor{M. Bittl} \institute{}
{\large \underline{Melanie Bittl}}\\
University of Erlangen-Nuremberg\\
\\ \vspace{4mm}{\large Dmitri Kuzmin}\\
University of Erlangen-Nuremberg\\
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}.
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