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Bubble finite elements for the primitive equations of the ocean. (English) Zbl 1095.76031
Summary: We introduce two original mixed methods for the numerical resolution of the (stationary) primitive equations (PE) of the ocean. The PE govern the behavior of oceanic flows in shallow domains for large time scales. We use a reduced formulation [J. L. Lions et al., Nonlinearity 5, No. 5, 1007–1053 (1992; Zbl 0766.35039)] involving horizontal velocities and surface pressures. By using bubble functions constructed ad-hoc, we are able to define two stable mixed methods requiring a low number of degrees of freedom. The first one is based on the addition of bubbles of reduced support to \(\mathbb P_1(\mathbf x) \otimes \mathbb P_1(z)\) velocities elementwise. The second one makes use of conic bubbles of extended support along the vertical coordinate. The latter constitutes a genuine mini-element for the PE, e.g., it requires the least number of extra degrees of freedom to stabilize piecewise linear hydrostatic pressures. Both methods satisfy a specific inf-sup condition and provide stability and convergence. Finally, we compare several numerical features of the proposed pairs in the context of other finite element methods found in the literature.

76M10 Finite element methods applied to problems in fluid mechanics
76U05 General theory of rotating fluids
86A05 Hydrology, hydrography, oceanography
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