A comparison of three finite elements to solve the linear shallow water equations

The purpose of the present study is to select a convenient mixed finite element formulation for ocean modelling. The finite element equivalents of Arakawaʹs A-, B- and C-grids are investigated by using the linear shallow water equations. Numerical and analytical techniques are used to study the types of computational noise present in each element. It is shown that the P1P1 and the P1P0 element (the equivalents of the A- and B-grids respectively) allow the presence of spurious computational modes in the elevation field. For the P1P1 element, these modes can be filtered out by adding a stabilizing term to the continuity equation. This method, although consistent, can lead to dissipative unphysical effects at the discrete level. The P1⊥P0 element or low order Raviart–Thomas element (corresponding to the C-grid) is free of elevation noise and represents well inertia-gravity waves when the deformation radius is resolved but presents computational velocity modes. These modes are however filtered out in a more complex model in which the momentum diffusion term is not neglected.