### 2001-02

#### The inf-sup Condition For The Mapped \$Q_k-P_{k-1}^{disc}\$ Element In Arbitrary Space Dimensions

Series: 2001-02, Preprints

MSC:
65N12 Stability and convergence of numerical methods
65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods

Abstract:
One of the most popular pairs of finite elements is the \$Q_k-P_{k-1}^{disc}\$
element for which two possible versions of the 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\$ or
define a mapped version where the pressure space is defined by a transformed
polynomial space on a reference cell. Since the reference transformation is
in general not affine but multilinear, the two variants are not equal. 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\ge 2\$ in any space dimension.

Keywords:
Babu\v{s}ka-Brezzi condition, Stokes problem, finite element method