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

摘要

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 analysed. The main result, which improves previous suboptimal estimates as those in Prohl & Ruzicka (2001, On fully implicit space-time discretization for motions of incompressible fluids with shear dependent viscosities: the case p <= 2. SIAM J. Numer. Anal., 39, 214-249) is the optimal O(k + h) error estimate valid in the range p is an element of (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 delta is an element of [0, delta(0)] of the extra stress tensor.