Numerical.tex

In this section we aim to verify the rates of convergence, developed in
\autoref{ch:Errors}, for both the QGE and SQGE. To this end we first use the
\emph{method of manufactured solutions} on a rectangular domain and calculate the
$L^2,\,H^1,\,H^2$ errors from the difference of the exact solution and the
numerical solution obtained from applying the Argyris element to the
corresponding problem. Additionally, the rates of convergence for the two-level
method applied to the SQGE is verified in a similar manner. Efficiency of the
two-level method is also demonstrated by looking at the time required to
simulate the SQGE in both the one-level method and the two-level method.

After verifying the rates of convergence obtained from the method of
manufactured solutions, we then look to studying the rates of convergence on
realistic domains with realistic parameter values. To this end we apply the FEM
to the SQGE on both the Mediterranean and North Atlantic Ocean. We treat the
finest mesh as our ``true solution'' and compare to the solutions obtained on
coarser meshes.