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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.

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