Digital Object Identifier (DOI) 10.1007/s00211010383 Numer. Math. (2002) 93: 201–221 Numerische Mathematik Convergence analysis and error estimates for a parallel algorithm for solving the Navier-Stokes equations I. Albarreal 1,⋆ , M.C. Calzada 2,⋆⋆ , J.L. Cruz 2,⋆⋆ , E. Fern ´ andez-Cara 1,⋆ , J. Galo 2,⋆⋆ , M. Mar´ ın 2,⋆⋆ 1 Department of Differential Equations and Numerical Analysis, University of Sevilla, Tarfia s/n, 41012 Sevilla, Spain; e-mail: cara@numer.us.es 2 Department of Computational Science and Numerical Analysis, University of C´ ordoba, Campus de Rabanales, Ed. C2-3, 14071 C ´ ordoba, Spain; e-mail: jlcruz@uco.es Received April 20, 2001 / Revised version received May 21, 2001 / Published online March 8, 2002 – c  Springer-Verlag Summary. This paper is concerned with the analysis of the convergence and the derivation of error estimates for a parallel algorithm which is used to solve the incompressible Navier-Stokes equations. As usual, the main idea is to split the main differential operator; this allows to consider independently the two main difficulties, namely nonlinearity and incompressibility. The results justify the observed accuracy of related numerical results. Mathematics Subject Classification (1991): 65M12 1 Introduction Let Ω ⊂ R d be a regular bounded domain (d =2 or 3) and assume that T> 0. In this paper, we will consider a fluid governed by the Navier-Stokes equations in Ω × (0,T ), that is: ∂u ∂t - ν∆u +(u ·∇)u + ∇p = f in Ω × (0,T ), ∇· u =0 in Ω × (0,T ). (1) Here, u = u(x,t) is the velocity field, p = p(x,t) is the pressure, ν> 0 is the kinematic viscosity (a constant) and f = f (x,t) is the density function ⋆ Partially supported by D.G.E.S. (Spain), Proyecto PB98–1134 ⋆⋆ Partially supported by D.G.E.S. (Spain), Proyecto PB96–0986