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

• Published in: Ocean modelling (Oxford) 2003, Vol.5 (1), p.17-35
• Main Authors: Hanert, E., Legat, V., Deleersnijder, E.
• Format: Article
• Language: English
• Subjects: Finite elements; Ocean modelling; Spurious computational modes; Unstructured grids
• ISSN: 1463-5003; 1463-5011
• DOI: 10.1016/S1463-5003(02)00012-4

The purpose of the present study is to select a convenient mixed finite element formulation for ocean modelling. The finite element equivalents of Arakawa's 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 P 1 P 1 and the P 1 P 0 element (the equivalents of the A- and B-grids respectively) allow the presence of spurious computational modes in the elevation field. For the P 1 P 1 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 P 1 ⊥ P 0 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.