Preconditioning mixed finite elements for tide models.

We describe finite element methods for the linearized rotating shallow water equations which govern tides. Symplectic Euler and Crank-Nicolson time-stepping strategies have good energy preservation properties, which is desirable for tide modeling, but require careful treatment of linear algebra. For symplectic Euler, we have to invert the Raviart-Thomas element mass matrix at every time step. Thus we give estimates for the eigenvalues of these mass matrices. Crank-Nicolson, being fully implicit, has a more complicated system of equations which requires inverting the entire system. For this, we present an effective block preconditioner using parameter-weighted norms in H(div). We give results that are nearly dependent of the given constants. Finally, we provide numerical results that confirm this theory.