Mixed finite elements for global tide models with nonlinear damping
Provides theoretical guarantees for numerical simulations of global tide models, which is incremental for computational geophysics.
The paper proves long-time stability and provides error estimates for mixed finite element methods applied to rotating shallow water equations with nonlinear drag, confirmed by numerical results.
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.