Preconditioning mixed finite elements for tide models.


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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.



Finite elements. Mixed methods. Preconditioning. Tide model. Rotating shallow water equations.