Errors.tex

The main goal of this Chapter is to develop rigorous numerical analysis for the
conforming FE discretization of the QGE \eqref{eqn:QGE_psi}. In particular we
focus on the use of the Argyris element, but the majority of the results apply
to any conforming FE discretization of QGE \eqref{eqn:QGE_psi}, SQGE
\eqref{eqn:SQGE_Psi}, and the two-level method applied to SQGE
\eqref{eqn:SQGE_Psi}.

In the sections that follow we rely heavily on the Young inequality, H\"older
inequality , Cauchy-Schwarz inequality, Poincar\'e-Friedrichs inequality, and
Ladyzhenskaya inequality, so for completeness we will state them here.

\begin{definition} \label{def:Young}
  \textbf{Young inequality} \cite{Royden2010}:\\
  For $1 < p,q < \infty$, $\dfrac{1}{p} + \dfrac{1}{q} = 1$, and any two
  positive numbers $a$ and $b$,
  \begin{equation}
    ab \le \frac{a^p}{p} + \frac{b^q}{q}.
    \label{eqn:Young}
  \end{equation}
\end{definition}
\begin{definition} \label{def:Holder}
  \textbf{H\"older inequality} \cite{Royden2010}:\\
  Let $\Omega$ be a measurable set, $1\le p,q < \infty$, and $\dfrac{1}{p} +
  \dfrac{1}{q} = 1$. If $f \in L^p(\Omega)$ and $g \in L^q(\Omega)$, then their
  product $f\, g$ is integrable over $\Omega$ and
  \begin{equation}
    \int_{\Omega}\! |f\, g| \, d\mathbf{x} \le \|f\|_p\, \|g\|_q.
    \label{eqn:Holder}
  \end{equation}
\end{definition}
\begin{definition} \label{def:Cauchy-Schwarz}
  \textbf{Cauchy-Schwarz inequality} \cite{Royden2010}:\\
  Let $\Omega$ be a measurable set and $f$ and $g$ be square-integrable over
  $\Omega$. Then their product $f\, g$ is also over $\Omega$ and
  \begin{equation}
    \int_{\Omega}\! |f\,g|\, d\mathbf{x} \le \sqrt{\int_{\Omega}\! f^2 \,
      d\mathbf{x}}\, \sqrt{\int_{\Omega}\! g^2 \, d\mathbf{x}}.
    \label{eqn:Cauchy}
  \end{equation}

\end{definition}
\begin{definition} \label{def:Poincare}
  \textbf{Poincar\'e-Friedrichs inequality} \cite{Layton08}:\\
  Let $\Omega$ be a measurable set and $u \in H^1_0(\Omega)$. Then there is a
  positive constant $C_P = C_P(\Omega)$ such that
  \begin{equation}
    \|u\| \le C_P \|\nabla u\| \quad \forall u \in X.
    \label{eqn:Poincare}
  \end{equation}
\end{definition}
\begin{definition} \label{def:Ladyzhenskaya}
  \textbf{Ladyzhenskaya inequality \cite{Layton08}:}\\
  For any vector function $u\,:\, \R^2 \rightarrow \R^2$ with compact support
  and with the indicated finite $L^4$ and $L^2$ norms,
  \begin{equation}
    \|u\|_{L^4(\R^2)} \le 2^{\nicefrac{1}{4}}
      \|u\|^{\nicefrac{1}{2}}_{L^2(\R^2)}
      \|\nabla u\|^{\nicefrac{1}{2}}_{L^2(\R^2)}.
    \label{eqn:Ladyzhenskaya}
  \end{equation}
\end{definition}
With these definitions we can now proceed with the error analysis.