# Mixed finite elements for global tide models with nonlinear damping

## TL;DR

This paper develops and analyzes mixed finite element methods for global tide models with nonlinear damping, demonstrating stability, damping rates, and convergence through theoretical proofs and numerical validation.

## Contribution

It introduces a new mixed finite element approach for tide models with nonlinear damping, providing stability analysis, error estimates, and numerical validation.

## Key findings

- Proven long-time stability without energy accumulation.
- Derived damping rates for unforced systems.
- Numerical results confirm theoretical energy damping and convergence.

## Abstract

We study mixed finite element methods for the rotating shallow water equations with linearized momentum terms but nonlinear drag. By means of an equivalent second-order formulation, we prove long-time stability of the system without energy accumulation. We also give rates of damping in unforced systems and various continuous dependence results on initial conditions and forcing terms. \emph{A priori} error estimates for the momentum and free surface elevation are given in $L^2$ as well as for the time derivative and divergence of the momentum. Numerical results confirm the theoretical results regarding both energy damping and convergence rates.