Framework.tex

In this chapter we build the mathematical framework needed to apply the FEM. To
this end, we introduce the functional spaces required for the formulation of the
QGE, SQGE, and we also discuss the well-posedness of SQGE. All of which are
prerequisites for both apply the FE discretization, as is done in
\autoref{ch:FEM}, and the error estimates proven in \autoref{ch:Errors}.

For completeness we will first define an inner product, $L^p$ spaces, and
Sobolev spaces along with their associated norms.

\begin{definition} \label{def:InnerProduct}
  \textbf{Inner Product} \cite{Kreyszig1989}: \\
  Let $f$ and $g$ be square integrable functions such that $f,g : \Omega
  \rightarrow \R$.  Then their inner product is defined as
  \begin{equation}
    (f,g) := \int_{\Omega}\! f \, g\, d\mathbf{x}
    \label{eqn:InnerProduct}
  \end{equation}
\end{definition}

\begin{definition} \label{def:LpNorm}
  \textbf{$L^p$-norm} \cite{Kreyszig1989}: \\
  Let $1\le p < \infty$ and $f \in L^p(\Omega)$ and $X$ be a normed-space, over
  $\Omega$, with the associated norm
  \begin{equation}
    \|f\|_{L^p(\Omega)} := \left(\int_{\Omega}\! |f|^p\, d\mathbf{x}\right)^{\frac{1}{p}}.
    \label{eqn:LpNorm}
  \end{equation}
  Then the completion of the norm-space $X$, denoted $L^p(\Omega)$ is called an
  $L^p$-space and we denote the norm associated with $L^p(\Omega)$ as
  $\|\cdot\|_{L^p(\Omega)}$ and is defined as in \eqref{eqn:LpNorm}.
\end{definition}
We will denote the typical $L^2$-norm by $\|\cdot\|$.

\begin{definition} \label{SobolevSpace}
  \textbf{Sobolev space} \cite{Evans1989}: \\
  Let $1\le p < \infty$, $f \in L^p(\Omega)$ be a real valued function, $s$ be
  non-negative, and $\alpha$ be a multi-index with $|\alpha| \le s$. Then the
  Sobolev space $W^{s,p}(\Omega)$ is defined as
  \begin{equation}
    W^{s,p}(\Omega) := \left\{ f\in L^p(\Omega) : D^{\alpha} f \in
      L^p(\Omega)\quad \forall |\alpha| \le s\right\},
    \label{eqn:Sobolev}
  \end{equation}
  where $D^{\alpha}f$ is the order $\alpha$ derivative of $f$ in the weak sense.
  The associated norm is given by
  \begin{equation}
    \|f\|_{W^{s,p}(\Omega)}^p := \sum_{|\alpha|\le s}
      \|D^{\alpha}f\|_{L^p(\Omega)}^p.
    \label{eqn:HkpNorm}
  \end{equation}
\end{definition}
In what follows we denote $W^{s,2}(\Omega)$ as $H^s(\Omega)$ and
$\|f\|_{H^s(\Omega)}$ as $\|f\|_s$, while the semi-norm will be denoted
as $|f|_s := \|D^s f\|$.