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On the stability of approximations for the Stokes problem using different finite element spaces for each component of the velocity. (English) Zbl 1329.76172
Summary: The stability of velocity and pressure mixed approximations of the Stokes problem is studied, when different finite element (FE) spaces for each component of the velocity field are considered. Using the macro-element technique of Stenberg, analytical results are obtained for some new combinations of FE with globally continuous and piecewise linear pressure. These new combinations are introduced with the idea of reducing the number of degrees of freedom in some of the velocity components. Although the resulting FE are not stable in general, we show their stability in a wide family of meshes (uniformly unstructured meshes). Moreover, this method can be extended to any mesh family whenever a post-processing be performed in order to convert it in an unstructured mesh. Some \(2D\) and \(3D\) numerical simulations are provided to agree with the previous analysis.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows
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