TwoLevelAlgorithm.tex

We consider an approximate solution to \eqref{eqn:SQGEWF} by a two-level finite
element procedure \cite{Fairag98,Layton93}. Let $X^h,\, X^H \subset
H^2_0(\Omega)$ denote two conforming finite element spaces with $H \gg h$. We
compute an approximate solution $\psi^h$ in the finite element space $X^h$ by
solving a linear system for the degrees of freedom in $X^h$.  This
linear system requires: first, the computation of the approximate solution $\psi^H$ to
the nonlinear system in the finite element space $X^H$, where the mesh is very
coarse, i.e. $H \gg h$, and then using this solution, $\psi^H$, to linearize the
problem on the fine mesh. This procedure is as follows:

\begin{algorithm}%[H]
  \caption{}%Two-Level algorithm for the Streamfunction formulation of QGE}
  \label{alg:TwoLevel}
  \begin{enumerate}[Step 1:]
    \item Solve the following nonlinear system on a coarse mesh for $\psi^H\in X^H$:
    \begin{equation}
      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 \text{for all } \chi^H \in X^H.
      \label{eqn:Coarse}
    \end{equation}
    \item Solve the following linear system on a fine mesh for $\psi^h\in X^h$:
    \begin{equation}
      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 \text{for all } \chi^h \in X^h.
      \label{eqn:Fine}
    \end{equation}
  \end{enumerate}
\end{algorithm}