How to save a bad element with weak boundary conditions

The P1–P1 finite element pair is known to allow the existence of spurious pressure (surface elevation) modes for the shallow water equations and to be unstable for mixed formulations. We show that this behavior is strongly influenced by the strong or the weak enforcement of the impermeability boundary conditions. A numerical analysis of the Stommel model is performed for both P1–P1 and P1NC–P1 mixed formulations. Steady and transient test cases are considered. We observe that the P1–P1 element exhibits stable discrete solutions with weak boundary conditions or with fully unstructured meshes.