Numerical Analysis

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 considered the QGE in the pure streamfunction formulation; this yields a fourth-order partial differential equation in the streamfunction. We developed a high order (C¹, Argyris) conforming finite element method for discretizations.