New error estimates for a viscosity-splitting scheme in time for the three-dimensional Navier-Stokes equations

Abstract

This work is devoted to obtaining optimal error estimates (for the velocity and the pressure) for a first-order time-discrete splitting scheme (using decomposition of the viscosity) for solving the incompressible time-dependent Navier-Stokes equations in three-dimensional domains. This scheme has been previously studied by other authors (Blasco et al. 1997 Int. J. Numer. Methods Fluids, 28, 1391-1419; Blasco & Codina, 2004, Appl. Numer. Math., 51, 1-17), but the main novelty of this paper is to establish optimal error estimates for the pressure. This behaviour has been numerically observed, but never hitherto proved. Moreover, owing to the introduction of a weight for the initial steps, these optimal error estimates are obtained without imposing either constraints on the time step or global compatibility conditions for the pressure at the initial time (related to further regularity hypotheses on the exact solution). © The author 2010. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.