seamlessocean.tex

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\title{Compatible finite element methods for geophysical fluid dynamics}
\author{Colin J. Cotter, David A. Ham, Jemma Shipton, Lawrence Mitchell, Andrew T.T. McRae}
\department{Department of Mathematics}
\institution{Imperial College London}
\email{colin.cotter@imperial.ac.uk}
\url{http://www.imperial.ac.uk/people/colin.cotter}
\def\MM#1{\boldsymbol{#1}}
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\group{The Gung Ho Project}
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% College Ocean Model.
\project{{Firedrake: http://firedrakeproject.org}}

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\begin{document}
\begin{multicols*}{3}
\section{Introduction}
Compatible finite elements form the basis of the Gung Ho dynamical
core formulation, extending C-grid staggering to pseudo-uniform meshes
that avoid the communication bottleneck caused by the poles in
a lat-long grid. They are also very suitable for
multiresolution meshes for seamless ocean modelling.

\section{What are compatible finite elements useful for?}
Same properties as C-grid \cite{cotter2012mixed}:
\vspace{-1cm}
\begin{multicols}{2}
\begin{packed_enum}
\item No spurious pressure modes,
\item No spurious inertial oscillations,
\item Steady geostrophic modes,
\item Conservation of linear energy.
\end{packed_enum}
\end{multicols}
\vspace{-0.5cm}
Plus:
\begin{packed_enum}
\item No orthogonality requirements on meshes; can use triangles or
  quadrilaterals.
\item Consistent discretisation of Coriolis, and higher-order options.
\item Flexibility to avoid spurious C-grid inertia-gravity modes on triangles \cite{danilov2010utility}.
\end{packed_enum}
\vspace{-1cm}
%This makes compatible finite element methods an attractive option for
%seamless ocean modelling on multiresolution unstructured meshes.

\section{What are compatible finite element methods?}

Sets/spaces of finite element functions are defined by:
\begin{packed_enum}
\item The choice of polynomials used in each cell (e.g., linear,
  quadratic, etc.),
\item The continuity constraints on the functions between neighbouring
  cells.
\end{packed_enum}
In the compatible finite element framework, we choose:
\begin{packed_enum}
  \item Continuity in normal component for velocity space $V_1$.
  \item Discontinuous functions for pressure space $V_2$.
    \item $\MM{u}\in V_1 \implies
\nabla\cdot\MM{u}\in V_2$.
\end{packed_enum}
\vspace{-0.5cm}
\begin{equation*}
  \begin{CD}
H({\rm div})     @>\nabla\cdot >>  L^2 \\
@VV\pi_1V        @VV\pi_2 V\\
V_1     @>\nabla\cdot>>  V_2
\end{CD}
\end{equation*}
\vspace{-1cm}
\section{Compatible spaces for rotating shallow water}
\parbox{25cm}{\hrule height 0pt width 0pt
  {\bfseries RT0-DG0 on quads.}
RT0: In reference element, $x$-component is linear in $x$, constant in
$y$; $y$ component is linear in $y$, constant in
$x$. DG0: constant functions.}
\hspace{1cm}
\parbox{5cm}
       {\hrule height 0pt width 0pt \includegraphics{RT0Q0DOFS}}
       \vspace{1cm} \\
\parbox{25cm}{\hrule height 0pt width 0pt {\bfseries RT1-DG1 on quads.}
RT1: In reference element, $x$-component is quadratic in $x$, linear in
$y$; $y$ component is quadratic in $y$, linear in
$x$. DG1: bilinear functions. Extension to higher order: RTk-DGk.}
\hspace{1cm}
\parbox{25cm}{\hrule height 0pt width 0pt \includegraphics{RT1Q1DOFS}}
\vspace{1cm} \\
\parbox{25cm}{\hrule height 0pt width 0pt {\bfseries BDM1-DG0 on triangles.}
  BDM1: Linear vector-valued functions. DG0: constant functions.}
\hspace{1cm}
\parbox{25cm}{\hrule height 0pt width 0pt 
\includegraphics{BDM1_DOFS}} \vspace{1cm} \\
\parbox{25cm}{\hrule height 0pt width 0pt {\bfseries BDM2-DG1 on triangles.}
BDM2: Quadratic vector-valued functions. DG1: Linear functions. Extension
to higher-order: BDMk-DG(k-1).}
\hspace{1cm}
\parbox{25cm}{\hrule height 0pt width 0pt \includegraphics{BDM2_DOFS}}
\vspace{1cm} \\
\parbox{25cm}{\hrule height 0pt width 0pt {\bfseries BDFM1-DG1 on triangles.}  BDFM1: Quadratic vector-valued functions
    with normal components constrained to be linear on edges. DG1:
    Linear functions.}
\hspace{1cm}
\parbox{25cm}{\hrule height 0pt width 0pt
  \includegraphics{BDFM1_DOFS}}

\section{Energy-enstrophy conserving discretisations for the nonlinear shallow water
  equations on the sphere}

Continuous finite element space $V_0$, with
$\psi \in V_0 \implies \nabla^\perp\psi \in V_1$, \cite{mcrae2014energy}
leads to 
energy-enstrophy conserving and energy conserving/enstrophy dissipating discretisations.
\\ \vspace{1cm}
%\includegraphics[width=15cm]{enenserrors}
  \raisebox{5cm}{\parbox{22cm}{
    \includegraphics[width=9cm]{uq_20_2}
    \includegraphics[width=9cm]{uq_30_2}\\
    \includegraphics[width=9cm]{uq_40_2}
    \includegraphics[width=9cm]{uq_50_2}}}
  \hspace{-3cm}
  \raisebox{5cm}{\parbox{22cm}{
    \includegraphics[width=17cm]{2.eps}\\
    \includegraphics[width=17cm]{4.eps}
    }}

  \section{3D discretisations: Hexahedral/prismatic elements formed from tensor products
  \cite{tensor2016}}
\centerline{\raisebox{1.8cm}{\bfseries Density space:} \hspace{1cm}
\includegraphics[width=4cm]{l2_tri} 
\raisebox{1cm}{\Large $\otimes$}
\raisebox{0.0cm}{\includegraphics[width=4cm,angle=90]{l2_int}} \raisebox{0.4cm}{\Large $=$}
\includegraphics[width=3.2cm]{l2l2_prism}}
\centerline{\raisebox{1.8cm}{\bfseries Horizontal part of velocity space:}\hspace{1cm}
\includegraphics[width=4cm]{hdiv_tri} 
 \raisebox{1cm}{\Large $\otimes$}
\raisebox{0.0cm}{\includegraphics[width=4cm,angle=90]{l2_int}} 
 \raisebox{1.2cm}{\Large $=$}
 \raisebox{0.4cm}{\includegraphics[width=3.2cm]{hdiv_h_prism}}}
\centerline{\raisebox{1.8cm}{\bfseries Vertical part of velocity space:}\hspace{1cm}
\includegraphics[width=4cm]{l2_tri} 
 \raisebox{1cm}{\Large $\otimes$}
 \raisebox{0.0cm}{\includegraphics[width=4cm,angle=90]{h1_int}} 
 \raisebox{1.2cm}{\Large $=$}
 \raisebox{1mm}{\includegraphics[width=3.2cm]{hdiv_v_prism}}}
\centerline{\raisebox{1.8cm}{\bfseries Temperature space:}\hspace{1cm}
\includegraphics[width=4cm]{l2_tri} 
 \raisebox{1cm}{\Large $\otimes$}
 \raisebox{0.0cm}{\includegraphics[width=4cm,angle=90]{h1_int}} 
 \raisebox{1.2cm}{\Large $=$}
 \raisebox{1mm}{\includegraphics[width=3.2cm]{l2h1_prism}}}
 \section{Vertical slice tests}
  \centerline{
    \includegraphics[width=25cm]{gw_nh}
    }\centerline{
    \includegraphics[width=25cm]{dc_filled}
}

  \section{Parallel scaling tests}
  \centerline{\includegraphics[width=25cm]{gmres-preonlyS-ref6-layers40}}
\small
\bibliography{seamlessocean}
\end{multicols*}
\end{document}