### 2001-02

by Matthies, G., Tobiska, L..

**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