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Analysis of the hydrostatic Stokes problem and finite-element approximation in unstructured meshes

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Abstract

The stability of velocity and pressure mixed finite-element approximations in general meshes of the hydrostatic Stokes problem is studied, where two “inf-sup” conditions appear associated to the two constraints of the problem; namely incompressibility and hydrostatic pressure. Since these two constraints have different properties, it is not easy to choose finite element spaces satisfying both. From the analytical point of view, two main results are established; the stability of an anisotropic approximation of the velocity (using different spaces for horizontal and vertical velocities) with piecewise constant pressures, and the unstability of standard (isotropic) approximations which are stable for the Stokes problem, like the mini-element or the Taylor-Hood element. Moreover, we give some numerical simulations, which agree with the previous analytical results and allow us to conjecture the stability of some anisotropic approximations of the velocity with continuous piecewise linear pressure in unstructured meshes.

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Authors and Affiliations

  1. Departamento EDAN and IMUS, Universidad de Sevilla, Aptdo. 1160, 41080, Sevilla, Spain

    F. Guillén-González

  2. Departamento de Matemáticas, Universidad de Cádiz, Polígono Río San Pedro S/N, Puerto Real, 11510, Cádiz, Spain

    J. R. Rodríguez-Galván

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  1. F. Guillén-González
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  2. J. R. Rodríguez-Galván
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Correspondence to F. Guillén-González.

Additional information

The first author has been partially supported by projects MTM2009-12927 and MTM2012-32325 (DGI-MEC, Spain) and the second one by the research group FQM-315 (Junta de Andalucía).

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Guillén-González, F., Rodríguez-Galván, J.R. Analysis of the hydrostatic Stokes problem and finite-element approximation in unstructured meshes. Numer. Math. 130, 225–256 (2015). https://doi.org/10.1007/s00211-014-0663-8

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  • DOI: https://doi.org/10.1007/s00211-014-0663-8

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