Mixed finite elements for global tide models

TL;DR

This paper develops and analyzes mixed finite element methods for linearized rotating shallow water equations, demonstrating long-term stability and convergence through theoretical proofs and numerical validation.

Contribution

It introduces a stable mixed finite element approach for tide modeling, providing rigorous error estimates and confirming stability and accuracy numerically.

Findings

Long-time stability without energy accumulation.

Proven error estimates in $L^2$ norms.

Numerical results confirm theoretical convergence and stability.

Abstract

We study mixed finite element methods for the linearized rotating shallow water equations with linear drag and forcing terms. By means of a strong energy estimate for an equivalent second-order formulation for the linearized momentum, we prove long-time stability of the system without energy accumulation -- the geotryptic state. A priori error estimates for the linearized momentum and free surface elevation are given in $L^2$ as well as for the time derivative and divergence of the linearized momentum. Numerical results confirm the theoretical results regarding both energy damping and convergence rates.