Analysis of the Yosida method for the incompressible Navier-Stokes equations

The Yosida method was introduced in (Quarteroni et al., to appear) for the numerical approximation of the incompressible unsteady Navier-Stokes equati ons. From the algebraic viewpoint, it can be regarded as an inexact factori zation of the matrix arising from the space and time discretization of the problem. However, its differential interpretation resides on an elliptic st abilization of the continuity equation through the Yosida regularization of the Laplacian (see (Brezis, 1983, Ciarlet and Lions, 1991)), The motivatio n of this method as well as an extensive numerical validation were given in (Quarteroni et al., to appear). In this paper we carry out the analysis of this scheme. In particular, we c onsider a first-order time advancing unsplit method. In the case of the Sto kes problem, we prove unconditional stability and moreover that the splitti ng error introduced by the Yosida scheme does not affect the overall accura cy of the solution, which remains linear with respect to the time step. Som e numerical experiments, for both the Stokes and Navier-Stokes equations, a re presented in order to substantiate our theoretical results. (C) Elsevier , Paris.