The finite element method is the main tool that we used in the numerical discretization
of the models that we developed, such as, the new POD-ROM closure models and the new QGE
AD model. Along the way, we addressed new mathematical and computational challenges.
To measure the error of the FEM approximation in the new POD-ROMs,
we derived a new POD inverse estimate:
where $v_r$ is the POD projection of the FEM solution,
$M_r$ is the POD mass matrix and $S_r$ is the POD stiffness matrix.
We used this estimate together with the FEM approximability property and the
POD projection error estimate to prove:
By balancing the terms in the error bound, we derived the optimal model parameter value
The FEM discretization of the QGE is a natural approach since it allows an easy treatment of complex boundaries (e.g., continents for the ocean or mountains for the atmosphere).
We developed low-order ($C^0$) finite elements discretization of the streamfunction-vorticity formulation of the QGE. A more efficient computational alternative is the pure streamfunction formulation; this yields a fourth-order partial differential equation in the streamfunction. We used the Argyris element (a $C^1$ finite element) for discretizations.