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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
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