Optimal Error Estimates for a Semi-Implicit Euler Scheme for Incompressible Fluids with Shear Dependent Viscosities

Certain rheological behaviors of fluids in engineering sciences are modeled by power law ansatz with $p\in(1,2]$. In the present paper a semi-implicit time discretization scheme for such fluids is proposed. The main result is the optimal $\mathcal{O}(k)$ error estimate, where $k$ is the time step size. Our results hold in the range $p\in(3/2,2]$ (in the three-dimensional setting) for strong solutions of the continuous problem, whose existence is guaranteed under appropriate assumptions on the data. The estimates are uniform with respect to the degeneracy parameter $\delta\in[0,\delta_0]$ of the extra stress tensor. Additional regularity properties of the solution of the discrete problem are proved.

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