Convergence Analysis of High Order Algebraic Fractional Step Schemes for Time-Dependent Stokes Equations.

In this paper we analyze the family of Yosida algebraic fractional step schemes proposed in [A. Quarteroni, F. Saleri, and A. Veneziani, Comput. Methods Appl. Mech. Engrg., 188 (2000), pp. 505–526], [F. Saleri and A. Veneziani, SIAM J. Numer. Anal., 43 (2005), pp. 174–194], and [P. Gervasio, F. Saleri, and A. Veneziani, J. Comput. Phys., 214 (2006), pp. 347–365] when applied to time-dependent Stokes equations. Under suitable regularity assumptions on the data, splitting error estimates both for velocity and pressure are established. In particular we analyze the first three methods of this family, providing, respectively, convergence (of the fractional step solution towards the numerical solution achieved without any operator splitting) of orders 3/2, 5/2, 7/2 for the velocity and 1, 2, 3 for the pressure. Moreover a general way to set up higher-order schemes is proposed. The present analysis is carried out when spectral element methods are employed for space discretization.