Preconditioners for nonsymmetric linear systems in domain decomposition applied to a coupled discretization of Navier-Stokes equations

Abstract

We will consider the linear system generated by a coupled discretization and linearization method for the Navier-Stokes equations. This method consists of a discretization of the momentum equations to obtain the velocities and pressure at the faces of a finite volume, in terms of the values of these variables at the grid points followed by the integration of the momentum and continuity equations in the finite volumes.This integration leads to equations where the values of the variables at the cell faces are to be replaced by the expressions obtained at the previous stage.The linear system to be solved at each nonlinear iteration connects values of velocities and pressure at each grid point in each equation. The coefficient matrix is large, nonsymmetric, sparse, with non-null entries on the diagonal. The characteristics of these linear systems indicate the use of nonstationary iterative methods, for instance preconditioned GMES, for their solution. The application of preconditioners based on the incomplete factorization of the system matrix will be analyzed.When nonoverlapping domain decomposition is used to parallelize this method, the system matrix at each outer iteration is in block-partitioned form. The solution of Ax=b is then equivalent to the solution of the Schur complement reduced system (corresponding to the solution on the interfaces separating subdomains) followed by the update of the solution in the subdomains. Incomplete factorization preconditioners applied to S (Schur complement) are very expensive as they need the explicit construction of S.The use of other type of preconditioners based on probing techniques or approximate inverses will be considered. In the first case the preconditioner M will be a narrow band approximation of S obtained by the use of matrix-vector products only. In the second case, approximate inverse techniques allow the approximation of S by a less expensive sparse matrix to be used as a preconditioner.