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A posteriori error estimators for mixed finite element approximation of some quasi-Newtonian flows. (English) Zbl 0770.76034
The authors consider the Stokes problem in a bounded domain $$\Omega\subset\mathbb{R}^ 2$$, i.e. $$-\text{div } T(u,p)=f$$ in $$\Omega$$, $$\text{div } u=0$$ in $$\Omega$$, $$u=0$$ on $$\partial\Omega$$, where the tensor $$T(u,p)$$ is given according to the Carreau model by $$T(u,p)=2\mu(D_{II}(u)) D(u)-pI$$. Here $D(u)=\bigl(\textstyle{1\over 2}(u^ i_{x_ j}+u^ j_{x_ i})\bigr)_{1\leq i, j\leq 2},\quad D_{II}(u)=\textstyle{1\over 2}\sum^ 2_{i,j=1} D_{ij}(u) D_{ij}(u)$ and the function $$\mu(z)$$ has the form $$\mu(z)=\mu_ 0(1+\lambda z)^{(\beta/2)-1}$$, $$(\mu_ 0>0$$, $$\lambda>0$$, $$1<\beta\leq 2)$$. For a finite element approximation of the above problem they develop an a posteriori error estimator which is determined by the jumps of the normal derivative and local residuals of the finite element solution.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76A05 Non-Newtonian fluids 65N15 Error bounds for boundary value problems involving PDEs