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.
Highlights
Finite element methods are attractive for modelling the world’s oceans since implemention with triangular cells provides a means to accurately represent coastlines and topography [36]
We choose to concentrate on the mimetic, or compatible, finite element spaces which were proposed for numerical weather prediction in [7]
We prove the following results which are useful to tidal modellers: 1. For the mixed finite element methods that we consider, the spatial semidiscretization has an attracting solution in the presence of time-varying forcing
Summary
Finite element methods are attractive for modelling the world’s oceans since implemention with triangular cells provides a means to accurately represent coastlines and topography [36]. In this paper we take a different angle, and study the behavior of discretizations of forced-dissipative rotating shallow-water equations, which are used for predicting global barotropic tides. We shall make use of the discrete Helmholtz decomposition in order to show that mixed finite element discretizations of the forced-dissipative linear rotating shallowwater equations have the correct long-time energy behavior. A useful tool for predicting tides are the rotating shallow water equations, which provide a model of the barotropic (i.e., depth-averaged) dynamics of the ocean. Holst and Stern [15] have demonstrated that finite element analysis on discretized manifolds can be handled as a variational crime We summarize these findings and include an appendix at the end demonstrating how to apply their techniques to our own case.
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