ABSTRACT
This paper is on the study of a Schur complement operator arisen from the full discretization of the time-dependent Navier-Stokes equations and some inverse, stability and approximation properties are established. These properties indicate that the Schur complement operator behaves, in some sense, like a Laplacian operator discretized on a mesh of size comparable to the square root of temporal step-size, and they are instrumental for studying effective iterative methods for solving the underlying algebraic systems. As an application, they will be used to analyze a two-grid preconditioner for the Schur complement operator by the framework of auxiliary space method and the resulting analysis appears to be more transparent than what is available in the literature.
