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

TL;DR

This paper establishes an optimal error estimate for semi-implicit space-time discretization of unsteady generalized Newtonian fluids with stress tensors of $(p, abla)$-structure, improving previous suboptimal results in three-dimensional periodic domains.

Contribution

It provides the first optimal $O(k+h)$ error estimate for the semi-implicit scheme in the specified $p$-range, extending results to 3D and uniform with respect to stress degeneracy.

Findings

Optimal $O(k+h)$ error estimate proven.

Results valid in 3D periodic domains.

Estimates are uniform over stress degeneracy parameter.

Abstract

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.