Comparing the Size of Finite Element Spaces in 3D for Fluid Equations
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For the numerical approximation of incompressible fluid flow in mixed finite element spaces an inf-sup condition needs to be satisfied. Finite element spaces with exact divergence constraints are particularly useful for pressure robust approximation. However, especially in 3D for piecewise polynomial finite elements the identification of the pressure space and the proof of inf-sup stability pose major challenges and their analysis is still far from complete. On general meshes inf-sup stability requires high polynomial degree of the Scott-Vogelius element. Alternatively, one may consider split methods using lower polynomial degrees on split meshes. There has been a trend to seek methods of low polynomial degree on certain split meshes such as the Alfeld or the Worsey-Farin split. The expectation has been that the higher-order schemes would be too costly in terms of the number of degrees of freedom. Thus we present an approach of how to compare the size of different finite-element spaces in 3D. The finite elements for all but one low-order method on a split mesh have larger dimension than the Scott-Vogelius element that uses polynomials of fourth order for the velocity on the original mesh. The only exception from this is the method that uses piecewise linears on the Worsey-Farin split mesh. Even for polynomial degree 6 the Scott-Vogelius element has only about 50% higher dimension than most of the low-order split methods. This allows for a comparison of the computational cost and points out important directions of future investigation.