Abstract
In this paper we prove optimal error estimates for solutions with natural regularity of the equations describing the unsteady motion of incompressible shear-thinning fluids. We consider a full space-time semi-implicit scheme for the discretization. The main novelty, with respect to previous results, is that we obtain the estimates directly without introducing intermediate semi-discrete problems, which enables the treatment of homogeneous Dirichlet boundary conditions.
Highlights
In this paper we study a space-time discretization of the unsteady system describing the motion of homogeneous, incompressible shear-thinning fluids under homogeneous Dirichlet boundary conditions
Before we formulate the main result, proving optimal convergence rates for the error between the solution u of the continuous problem (1.1) and the discrete solution fumh gMm1⁄40 of the space-time discretization (1.4), we discuss the existence of solutions of (1.1) and
(iii) In [23, 24] the convergence of a fully implicit space-time discretization of the problem (1.1) in the case of homogeneous Dirichlet boundary conditions is proved for p[
Summary
In this paper we study a space-time discretization of the unsteady system describing the motion of homogeneous (for simplicity the density q is set equal to 1), incompressible shear-thinning fluids under homogeneous Dirichlet boundary conditions. 2.4) for solutions possessing a natural regularity, extending the results in [5] to the case of homogeneous Dirichlet boundary conditions. Our method differs from most previous investigations in as much as we use no intermediate semidiscrete problems to prove our result. We restrict ourselves to the three-dimensional setting, all results can be adapted to the general setting in d-dimensions. This article is part of the topical collection dedicated to In Honor of Professor Hideo Kozono’s 60th Birthday edited by Kazuhiro Ishige, Tohru Ozawa, Senjo Shimizu, and Yasushi Taniuchi
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