Preconditioning Techniques for Newton's Method for the Incompressible Navier–Stokes Equations

Abstract

Newton's method for the incompressible Navier—Stokes equations gives rise to large sparse non-symmetric indefinite matrices with a so-called saddle-point structure for which Schur complement preconditioners have proven to be effective when coupled with iterative methods of Krylov type. In this work we investigate the performance of two preconditioning techniques introduced originally for the Picard method for which both proved significantly superior to other approaches such as the Uzawa method. The first is a block preconditioner which is based on the algebraic structure of the system matrix. The other approach uses also a block preconditioner which is derived by considering the underlying partial differential operator matrix. Analysis and numerical comparison of the methods are presented.