Stabilised finite element methods for the Oseen problem on anisotropic quadrilateral meshes

Barrenechea, Gabriel R.; Wachtel, Andreas

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 99-122.

In this work we present and analyse new inf-sup stable, and stabilised, finite element methods for the Oseen equation in anisotropic quadrilateral meshes. The meshes are formed of closed parallelograms, and the analysis is restricted to two space dimensions. Starting with the lowest order ${\mathbb{Q}}_1^2\times {\mathbb{P}}_0$ pair, we first identify the pressure components that make this finite element pair to be non-inf-sup stable, especially with respect to the aspect ratio. We then propose a way to penalise them, both strongly, by directly removing them from the space, and weakly, by adding a stabilisation term based on jumps of the pressure across selected edges. Concerning the velocity stabilisation, we propose an enhanced grad-div term. Stability and optimal a priori error estimates are given, and the results are confirmed numerically.

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