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\begin{document}
%% Title Slide
\title{
The Dual-Wind Discontinuous Galerkin Method}
\author{\color{UNCGblue} \large
Tom Lewis}
\date{
SIAM Central States Sectional Meeting \\
Spring 2015}
%% Footer Text
\footertext{\hspace{7pt}
Tom Lewis
\hspace{25pt}
Email: \hspace{1pt} {\tt tllewis3@uncg.edu}
\hspace{25pt}
DWDG %%%% Short Talk Title
}
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\makeTitleSlide
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\begin{frame}{Outline}
\tableofcontents
\bigskip
{\scriptsize Collaborators:\\
\vspace{0.1in}
Michael Neilan, University of Pittsburgh \\
Wenqiang Feng, University of Tennessee \\
Steven Wise, University of Tennessee \\
}
\vfill
{\footnotesize \color{GCgreen} Supported in part by NSF}
\vfill
\end{frame}
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\section{Introduction}
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\begin{frame}{Overview}
\vspace{-0.2in}
{\bf PDE Problem:}
\begin{alignat*}{2}
- \Delta u & = f \qquad && \text{in } \Omega \subset \mathbb{R}^d , \\
u & = g \qquad && \text{on } \partial \Omega .
\end{alignat*}
\bigskip
{\bf Goal:}
Develop an optimally convergent DG method that:
\begin{itemize}
\item
is symmetric when written in primal form
\item
naturally enforces BCs without boundary penalization terms
\item
does not require interior stabilization
\end{itemize}
\bigskip
{\color{UTorange}
All of the following has been extended for Neumann BCs
and the biharmonic equation with Lagrange basis functions.}
\end{frame}
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\section{DG Derivative Operators}
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\begin{frame}{Interior Trace Values}
\vspace{-0.2in}
\hh
\begin{tikzpicture}[scale=0.75, transform shape]
\draw (0,0) -- (4,0) -- (0,4) -- cycle;
\draw (4,4) -- (0,4) -- (4,0) -- cycle;
\draw[thick] (0,4) -- (4,0);
\node[above left,blue] at (2.5,2) {$T^+_x$};
\node[below right,blue] at (2.4,2) {$T^+_y$};
\draw[->,blue,very thick,dashed] (1,3) -- (1,3.5);
\draw[->,blue,very thick,dashed] (1,3) -- (1.5,3);
\draw[->,thick] (1,3) -- (1.5,3.5);
\draw[->,thick] (1,3) -- (0.5,2.5);
\end{tikzpicture}
\hh
\begin{tikzpicture}[scale=0.75, transform shape]
\draw (0,0) -- (4,0) -- (0,4) -- cycle;
\draw (4,4) -- (0,4) -- (4,0) -- cycle;
\draw[thick] (0,4) -- (4,0);
\node[above left,red] at (1.6,2) {$T^-_x$};
\node[below right,red] at (1.5,2) {$T^-_y$};
\draw[->,red,very thick,dashed] (3,1) -- (3,0.5);
\draw[->,red,very thick,dashed] (3,1) -- (2.5,1);
\draw[->,thick] (3,1) -- (3.5,1.5);
\draw[->,thick] (3,1) -- (2.5,0.5);
\end{tikzpicture}
\hh
\bigskip
\hh
\begin{tikzpicture}[scale=0.75, transform shape]
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\draw (4,4) -- (0,4) -- (0,0) -- cycle;
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\node[below right,blue] at (1.75,2) {$T^+_x$};
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\draw[->,dashed] (3.5,3) -- (3.5,2.5);
\end{tikzpicture}
\hh
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\draw (0,0) -- (4,0) -- (4,4) -- cycle;
\draw (4,4) -- (0,4) -- (0,0) -- cycle;
\draw[thick] (0,0) -- (4,4);
\node[above left,red] at (2.25,2) {$T^-_x$};
\node[below right,red] at (1.75,2) {$T^-_y$};
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\draw[->,thick] (1,1) -- (1.5,0.5);
\draw[->,dashed] (0.5,1) -- (0.5,1.5);
\draw[->,dashed] (1,0.5) -- (1.5,0.5);
\end{tikzpicture}
\hh
\end{frame}
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\section{The DWDG Method}
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\begin{frame}{Convergence Results}
\vspace{-0.3in}
\begin{shaded}
Suppose that $\gamma_{\min}>0$.
Let $u_h\in V_{h,r}$ be the unique solution for the DWDG method
and $u\in H^{s+1}(\Omega)$ be the PDE solution with $1\leq s\leq r$.
Then there holds
\begin{align*}
\|u-u_h\|_{1,h}\leq C\Big(\sqrt{\gamma_{\max}}+\frac{1}{\sqrt{\gamma_{\min}}}\Big)h^{s}|u|_{H^{s+1}(\Omega)}.
\end{align*}
%
{\color{UTorange}
Moreover, if the triangulation is quasi-uniform, there exists a constant $C_* > 0$
such that, for $\gamma_{\min}>-C_*$,
there holds
\begin{align*}
\|u-u_h\|_{1,h}\leq C\Big(\sqrt{|\gamma_{\max}|}+\frac{1}{\sqrt{\gamma_{\min}+C_*}}\Big)h^{s}|u|_{H^{s+1}(\Omega)}.
\end{align*}
}
\end{shaded}
\end{frame}
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\section{Numerical Tests}
%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Test 3: $u = x^3 - 2 x y^2 + xy - y^2 + y - 3$
with a Cartesian mesh}
{\tiny
\begin{table}
\begin{tabular}{lllllllll}
&$h_x = h_y$ & $\|u-u_h\|_{L^2}$ & rate & $\|\nabla u-\nabla^+_h u_h\|_{L^2}$ & rate & $\|\nabla u-\nabla^-_h u_h\|_{L^2}$ & rate\smallskip\\
\hline
$r=0$
&1/4 &1.21E+00 & &3.01E+00 & &2.98E+00 &\\
&1/8 &6.32E-01 &0.93 &1.83E+00 &0.72 &1.86E+00 &0.68\\
&1/16 &3.07E-01 &1.04 &1.03E+00 &0.84 &1.06E+00 &0.82\\
&1/32 &1.48E-01 &1.06 &5.55E-01 &0.89 &5.73E-01 &0.88\\
\hline
$r=1$
&1/4 &1.81E-01 & &7.18E-01 & &7.89E-01 &\\
&1/8 &4.42E-02 &2.04 &4.88E-01 &0.56 &4.96E-01 &0.67\\
&1/16 &1.09E-02 &2.02 &2.86E-01 &0.77 &2.87E-01 &0.79\\
&1/32 &2.73E-03 &1.99 &1.54E-01 &0.89 &1.55E-01 &0.89\\
\hline
$r=2$
&1/4 &1.31E-02 & &1.39E-01 & &1.39E-01 &\\
&1/8 &1.56E-03 &3.07 &4.03E-02 &1.78 &4.03E-02 &1.78\\
&1/16 &1.92E-04 &3.03 &1.07E-02 &1.92 &1.07E-02 &1.92\\
&1/32 &2.37E-05 &3.01 &2.75E-03 &1.96 &2.75E-03 &1.96 \\
\hline
$r=3$
&1/8 &1.35E-10 & &2.23E-09 & &2.18E-09 &
\end{tabular}
\end{table}
}
\begin{center}
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\end{center}
\end{frame}
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\section{Conclusion}
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\begin{frame}{Summary}
\vspace{-0.2in}
{\small
\begin{itemize}
\item
We propose a symmetric DG method for Poisson's equation that is well-posed and exhibits
optimal convergence rates without interior or boundary penalization.
Dirichlet (and Neumann) boundary data appears naturally in the scheme.
\smallskip
\item
For piecewise constants on a Cartesian mesh, the method is equivalent to the standard
second order Finite Difference method.
\smallskip
\item
The method is based on combining and composing various DG FE derivative operators
similar to the construction of FD methods.
\smallskip
\item
The method utilizes multiple trace values by incorporating 2 discrete derivatives that
both incorporate $d$ trace values.
\smallskip
\item
The method can be extended for other second order elliptic problems, different boundary conditions,
and higher order problems.
\end{itemize}
}
\end{frame}
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\begin{frame}{References}
\vspace{-0.2in}
{\scriptsize
\begin{itemize}
\item
{\sc T.~Lewis and M.~Neilan},
{\em Convergence analysis of a symmetric dual-wind discontinuous Galerkin method},
J. Sci. Comput. Volume 59, Issue 3, p. 602 -- 625. 2014.
\medskip
\item
{\sc W. Feng, T. Lewis, and S. Wise},
{\em Discontinuous Galerkin derivative operators with applications to
second order elliptic problems and stability}.
Mathematical Meth. in App. Sciences, 2015 ({\em In Press}).
\medskip
\item
{\sc X.~Feng, T.~Lewis, and M.~Neilan},
{\em Discontinuous Galerkin
finite element differential calculus and applications to numerical solutions
of linear and nonlinear partial differential equations},
submitted. arXiv:1302.6984 [math.NA].
\medskip
\item
{\sc B.~Cockburn and B.~Dong},
{\em An analysis of the minimal dissipation local discontinuous
Galerkin method for convection-diffusion problems},
J. Sci. Comput., 32(2):233--262, 2007.
\end{itemize}
}
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\centerline{\color{UNCGyellow} \LARGE Thank you for your attention!}
\medskip
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