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ABSTRACT:
A Fortin operator is constructed to verify the discrete inf-sup condition for $P^2_0-P^1$ Taylor-Hood element and its variant $P^2_0-(P^1 + P_0)$ in two dimensions. The approach is closely related to the recent work by Mardal, Schoberl and Winther (Numer. Math. 2012). That is based on the isomorphism of the tangential edge bubble function space to a subspace of the lowest order edge element space. A more precise characterization of this subspace and a numerical quadrature are introduced to simplify the analysis and remove the mesh restriction. The constructed Fortin operator for $P^2_0-P^1$ element is uniformly bounded in both $H^1$ and $L^2$ norm for general shape regular triangulations.