01-02
The inf-sup Condition For The Mapped $Q_k-P_{k-1}^{disc}$ Element In Arbitrary Space Dimensions
Preprint series: 01-02, 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
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