conclusions.tex

In this thesis we studied the FE discretization of the pure streamfunction form
of the QGE. The pure streamfunction form of the QGE requires $C^1$ FEs to be
conforming, thus the implementation of the of the $C^1$ FE was discussed in
length. In particular, the implementation of the Argyris element was discussed
in \autoref{sec:Argyris}, including an in depth discussion of the novel
transformation developed by Dominguez et. el.  \cite{Dominguez08} in
\autoref{sse:Trans}. Additionally, we proposed a two-level FE discretization for
the SQGE in \autoref{sec:TwoLevel} based on the conforming FE discretization
used for the SQGE in \autoref{sec:SQGEFEM}. The two-level algorithm consists of
two steps. In the first step, the nonlinear system is solved on a coarse mesh.
In the second step, the nonlinear system is linearized around the approximation
found in the first step, and the resulting linear system is solved on the fine
mesh.  This method significantly improved simulation times for the SQGE over
that of the one-level method. Next, a conforming FE semi-discretization was
developed for the time-dependent QGE based upon the FE discretization developed
in \autoref{sec:SQGEFEM}.

For the conforming FE discretization of SQGE, in \autoref{sec:SQGEErrors} we
proved, for the first time, optimal error estimates in the $H^2,H^1$, and $L^2$
norms. The error analysis only relied on the FE discretization being conforming
and was not specific to any particular element. Specifically, for the Argyris
element we showed that the order of convergence in the $H^2,H^1,L^2$ norms were
$O(h^4),O(h^5),$ and $O(h^6)$, respectively. The error analysis in
\autoref{sec:SQGEErrors} was then used to develop new rigorous error estimates
for the two-level FE discretization. These estimates are optimal in the
following sense: for an appropriately chosen scaling between the coarse mesh,
$H$, and the fine mesh, $h$, the error in the two-level method is of the same
order as the error in the standard one-level method (i.e. solving the nonlinear
system directly on the fine mesh). For the semi-discretization of the QGE we
proved similar optimal error estimates, i.e. the order of convergence for the
energy norm is of $O(h^4)$.

Numerical experiments for the SQGE (\autoref{sec:SQGETests}), two-level
algorithm (\autoref{sec:SQGE2LTests}), and QGE (\autoref{sec:QGETests}) with the
Argyris element, were also carried out. For the SQGE a few benchmark problems
were used to verify our code. These benchmark problems included the Linear
Stommel model \cite{Vallis06,Myers}, and the Linear Stommel-Munk model
\cite{Cascon}. With the agreement of our solutions to that of the literature our
code was thus verfied. We then verified numerically the theoretical error
estimates developed in \autoref{sec:SQGETests} and performed numerical
experiments on non-rectangular domains, including a FE discretization of the
SQGE applied to the Mediterranean. The test results for the Mediterranean appeared
to be in agreement with observed surface currents and the numerical experiments
developed by Galan del Sastre \cite{Galan-del-Sastre2004}. The code which was
verified in \autoref{sec:SQGETests} was then modified to allow for the use of
the two-level method and numerical experiments were then carried out for the
two-level method.  These numerical experiments verified numerically the
theoretical error estimates developed in \autoref{sse:SQGE2LE}, both with
respect to the coarse mesh size, $H$, and the fine mesh size, $h$.  Furthermore,
the numerical results showed that, for an appropriate scaling between the coarse
and fine meshes, the two-level method significantly decreases the computational
time of the standard one-level method. Next, the same code which was developed
and verified for the SQGE was then modified to deal with time-dependence. We
applied a implicit Euler scheme and verified numerically the theoretical spatial
rates of convergence developed for the semi-discretization. Additionally, we
observed expected rates of convergence in time for the implicit Euler scheme
($O(h)$).

The QGE has many unique challenges for numerical modelling. These challenges
include, but are not limited to, unstable solutions, resulting from internal
layers and western boundary layers, and high computational cost for large
domains, such as the North Atlantic. To address these issues we plan to extend
these studies in several directions, including stabilization methods such as
\emph{Petrov-Galerkin} stabilization, \emph{adaptive mesh refinement}, model
reduction using \emph{Proper Orthogonal Decomposition}, and the incorporation of
observed windstress data, which will include \emph{parameter estimation}.
Additionally, the streamfunction formulation suffers from non-unique
streamfunctions for multiply connected domains, such as the ocean, seas, and
large lakes with islands. To address this issue a method such as the one
developed by van Gijzen et. al. \cite{van-Gijzen1998} will be explored.  Thus, a
more realistic ocean mesh will be used and islands will no longer have to be
connected to continents.