Development of a fully coupled control‐volume finite element method for the incompressible Navier–Stokes equations

Abstract

AbstractThis paper proposes and investigates fully coupled control‐volume finite element method (CVFEM) for solving the two‐dimensional incompressible Navier–Stokes equations. The proposed method borrows many of its features from the segregated CVFEM described by Baliga et al. Thus finite‐volume discretization is employed on a colocated grid using either the MAW or the FLO schemes and an element‐by‐element assembling procedure is applied for the construction of the discretizations equations. In this paper, and unlike the case for most fully coupled formulations available in the literature, the Poisson pressure equation has been retained from the segregated approach. The use of a pressure equation leads to an unfavourable size increase of the fully coupled linear system, but significantly improves the system's conditioning. The fully coupled system obtained is solved using an ILUT preconditioned GMRES algorithm. The other important element in this paper is the proposal of a Newton linearization of the convection terms in lieu of the common Picard iteration procedure. A systematic comparison between two segregated and four fully coupled fomulations has been presented which has allowed for an evaluation of the individual benefits and strengths of the coupling and linearization procedure by studying lid‐driven cavity problems and flows past a circular cylinder. All coupled formulations have proven to be significantly superior both in robustness and efficiency, as compared with the segregated formulation. In some circumstances, the coupled methods yield a converged solution of the system of discretized equations constructed using the FLO scheme, while the segregated formulations diverge. Compared to Picard's linearization, Newton's linearization is more efficient at reducing the number of iterations needed to converge, but requires more computational effort per iteration from the linear equation solver. Furthermore, the Jacobian matrix should include contributions from the nonlinearity appearing at both the governing‐equation level and the interpolation‐scheme level to ensure Newton's method convergence. The key element in guaranteeing successful, fully coupled solutions lies in the use of an efficient linear equation solver and preconditioner. Copyright © 2004 John Wiley & Sons, Ltd.