Stability of high-order Scott-Vogelius elements for 2D non-Newtonian incompressible flow

We consider the stability of high-order Scott-Vogelius elements for 2D non-Newtonian incompressible flow problems. For elements of degree 4 or higher, we construct a right-inverse of the divergence operator that is stable uniformly in the polynomial degree N N from L p L^{p} to W 1 , p \boldsymbol {W}^{1,p} , show that the associated inf-sup constant is bounded below by a constant that decays at worst like N − 3 | 1 2 − 1 p | N^{-3\left | \frac {1}{2} - \frac {1}{p}\right |} , and construct local Fortin operators with stability constants explicit in the polynomial degree. We demonstrate these results with several numerical examples suggesting that the p p -version method can offer superior convergence rates over the h h -version method even in the non-Newtonian setting.