SQGE.FEF.tex

In this section, we present the functional setting and some auxiliary results
for the FE discretization of the streamfunction formulation of the SQGE
\eqref{eqn:SQGEWF}. Let $\mathcal{T}^h$ denote a finite element triangulation of
$\Omega$ with mesh size (maximum triangle diameter) $h$. We consider a
\emph{conforming} FE discretization of \eqref{eqn:SQGEWF}, i.e., $X^h \subset X
= H_0^2(\Omega)$.

The FE discretization of the streamfunction formulation of the SQGE
\eqref{eqn:SQGEWF} reads:
\begin{equation}
  \begin{split}
    &\text{Find } \psi^h \in X^h \text{ such that} \\
    Re^{-1}(\Delta \psi^h, \Delta \chi^h)
      + b(\psi^h,\psi^h,&\chi^h)
      - Ro^{-1} (\psi_x^h,\chi^h)
      = Ro^{-1}(F,\chi^h),\quad \forall \, \chi^h \in X^h.
    \label{eqn:SQGEFEF}
  \end{split}
\end{equation}
Using standard arguments \cite{Girault79,Girault86}, one can prove that, if the
small data condition used in proving the well-posedness result for the
continuous case holds, then \eqref{eqn:SQGEFEF} has a unique solution $\psi^h$
(see Theorem 2.1 in \cite{Cayco86}). Furthermore, one can prove the following
stability result for $\psi^h$ using the same arguments as those used in the
proof of \eqref{thm:stability_sqge} for the continuous setting.
\begin{thm} \label{thm:stability_fem_sqge} The
  solution $\psi^h$ of \eqref{eqn:SQGEFEF} satisfies the following stability estimate:
 \begin{equation}
   |\psi^h|_2 \le Re \, Ro^{-1} \, \| F \|_{-2} .
   \label{eqn:stability_fem_sqge}
 \end{equation}
\end{thm}
\begin{proof}
  The proof is almost identical to the proof of \autoref{thm:stability_sqge},
  but is given here for completeness.

  Let $\chi^h = \psi^h$ in \eqref{eqn:SQGEFEF} which gives
  \begin{equation*}
    Re^{-1}(\Delta \psi^h, \Delta \psi^h)
      + b(\psi^h,\psi^h,\psi^h)
      - Ro^{-1} (\psi_x^h,\psi^h)
      = Ro^{-1} (F,\psi^h)\qquad  \forall \pi^h \in X^h.
  \end{equation*}
  Since, $b(\psi^h, \psi^h, \psi^h) =0$ and $(\psi_x^h,\psi^h)=0$ we have
  \begin{align*}
    Re^{-1}(\Delta \psi^h, \Delta \psi^h) &= Ro^{-1} (F,\psi^h) \\
    Re^{-1}\, \|\psi^h\|_2^2 &= Ro^{-1}\, (F,\psi^h) \\
    \|\psi^h\|_2 &\le Re\, Ro^{-1}\,\sup_{\psi^h \in X^h} \frac{(F,\psi^h)}{|\psi^h|_2} \\
    \|\psi^h\|_2 &\le Re\, Ro^{-1}\, \|F\|_{-2}.
  \end{align*}
  Thus, the proof is complete.
\end{proof}

Again, we point out that as noted in Section 6.1 in \cite{Ciarlet} (see also
Section 13.2 in \cite{Gunzburger89}, Section 3.1 in \cite{Johnson}, and Theorem
5.2 in \cite{Braess}), in order to develop a conforming FE discretization for the
SQGE \eqref{eqn:SQGEWF}, we are faced with the problem of constructing subspaces
of the space $H^2_0(\Omega)$.  As was discussed previously the Argyris finite
element is an element of class $C^1$ and therefore will be the FE used in this
thesis for the discretization of the SQGE.