The Inf-Sup Condition for the Mapped Q _k −P _k−1 ^disc Element in Arbitrary Space Dimensions

Abstract

 One of the most popular pairs of finite elements for solving mixed formulations of the Stokes and Navier–Stokes problem is the Q _k −P _k−1 ^disc element. Two possible versions of the discontinuous pressure space can be considered: one can either use an unmapped version of the P _k−1 ^disc space consisting of piecewise polynomial functions of degree at most k−1 on each cell or define a mapped version where the pressure space is defined as the image of a polynomial space on a reference cell. Since the reference transformation is in general not affine but multilinear, the two variants are not equal on arbitrary meshes. It is well-known, that the inf-sup condition is satisfied for the first variant. In the present paper we show that the latter approach satisfies the inf-sup condition as well for k≥2 in any space dimension.