Preconditioning mixed finite elements for tide models

TL;DR

This paper introduces a preconditioning technique for mixed finite element methods applied to tide models, improving computational efficiency while maintaining stability and energy conservation in simulations.

Contribution

It develops nearly robust weighted-norm preconditioners for the linear algebra systems in tide modeling, enhancing solver performance for discretized shallow water equations.

Findings

Preconditioners are nearly robust across physical and discretization parameters.

Numerical experiments confirm theoretical stability and efficiency.

The method supports energy conservation in Crank-Nicolson time-stepping.

Abstract

We describe a fully discrete mixed finite element method for the linearized rotating shallow water model, possibly with damping. While Crank-Nicolson time-stepping conserves energy in the absence of drag or forcing terms and is not subject to a CFL-like stability condition, it requires the inversion of a linear system at each step. We develop weighted-norm preconditioners for this algebraic system that are nearly robust with respect to the physical and discretization parameters in the system. Numerical experiments using Firedrake support the theoretical results.