Optimal error estimate for semi-implicit space-time discretization for the equations describing incompressible generalized Newtonian fluids

Berselli LC, Diening L, Růžička M (2015)
IMA Journal of Numerical Analysis 35(2): 680-697.

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In this paper we study the numerical error arising in the space-time approximation of unsteady generalized Newtonian fluids which possess a stress-tensor with $(p,\delta)$-structure. A semi-implicit time-discretization scheme coupled with conforming inf-sup stable finite element space discretization is analyzed. The main result, which improves previous suboptimal estimates as those in [A. Prohl, and M. Ruzicka, SIAM J. Numer. Anal., 39 (2001), pp. 214--249] is the optimal $O(k+h)$ error-estimate valid in the range $p\in (3/2,2]$, where $k$ and $h$ are the time-step and the mesh-size, respectively. Our results hold in three-dimensional domains (with periodic boundary conditions) and are uniform with respect to the degeneracy parameter $\in [0,\delta_0]$ of the extra stress tensor.
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Berselli LC, Diening L, Růžička M. Optimal error estimate for semi-implicit space-time discretization for the equations describing incompressible generalized Newtonian fluids. IMA Journal of Numerical Analysis. 2015;35(2):680-697.
Berselli, L. C., Diening, L., & Růžička, M. (2015). Optimal error estimate for semi-implicit space-time discretization for the equations describing incompressible generalized Newtonian fluids. IMA Journal of Numerical Analysis, 35(2), 680-697. doi:10.1093/imanum/dru008
Berselli, L. C., Diening, L., and Růžička, M. (2015). Optimal error estimate for semi-implicit space-time discretization for the equations describing incompressible generalized Newtonian fluids. IMA Journal of Numerical Analysis 35, 680-697.
Berselli, L.C., Diening, L., & Růžička, M., 2015. Optimal error estimate for semi-implicit space-time discretization for the equations describing incompressible generalized Newtonian fluids. IMA Journal of Numerical Analysis, 35(2), p 680-697.
L.C. Berselli, L. Diening, and M. Růžička, “Optimal error estimate for semi-implicit space-time discretization for the equations describing incompressible generalized Newtonian fluids”, IMA Journal of Numerical Analysis, vol. 35, 2015, pp. 680-697.
Berselli, L.C., Diening, L., Růžička, M.: Optimal error estimate for semi-implicit space-time discretization for the equations describing incompressible generalized Newtonian fluids. IMA Journal of Numerical Analysis. 35, 680-697 (2015).
Berselli, Luigi C., Diening, Lars, and Růžička, Michael. “Optimal error estimate for semi-implicit space-time discretization for the equations describing incompressible generalized Newtonian fluids”. IMA Journal of Numerical Analysis 35.2 (2015): 680-697.
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