# FEniCS Application to a Finite Element Quasi-Geostrophic Model. Linear and non-linear analysis of Munk-like Solutions and Data Assimilation Implementation Through Adjoint Method

## TL;DR

This paper develops a finite element model for the quasi-geostrophic equation using FEniCS, compares it with Navier-Stokes, and explores data assimilation via control theory and adjoint methods for large-scale problems.

## Contribution

It introduces a finite element approach for quasi-geostrophic equations with FEniCS and demonstrates the application of control theory and adjoint methods for data assimilation.

## Key findings

- Finite element approximation of quasi-geostrophic model successfully implemented.
- Comparison shows differences between Navier-Stokes and quasi-geostrophic models.
- Data assimilation approach via adjoint method enhances model accuracy.

## Abstract

The first aim of this work is to construct a Finite Elements model for quasi-geostrophic equation. This model is analyzed through FEniCS interface. As a second purpose, it shows the potential of the control theory to treat Data Assimilation large scale problems. In section one the finite element approximation of the quasi-geostrophic model is described. In section two some numerical results on different domains are shown. In section three a comparison between Navier Stokes model and quasi-geostrophic one is discussed. Last section is dedicated to a more complex data assimilation/control problem based on quasi-geostrophic equations, to test possible developments and improvements.