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In this work, three stabilized finite volume iterative schemes for the stationary Navier–Stokes equations are considered. Under the finite volume discretization at each iterative step, the iterative scheme I consists in solving the steady Stokes problem, iterative scheme II consists in solving the stationary linearized Navier–Stokes equations and iterative scheme III consists in solving the steady Oseen equations, respectively. We discuss the stabilities and convergence of three iterative methods. The iterative schemes I and II are stable and convergent under some strong uniqueness conditions, while iterative scheme III is unconditionally stable and convergent under the uniqueness condition. Finally, some numerical results are presented to verify the performance of these iterative schemes. [ABSTRACT FROM AUTHOR]

Abstract: In this work, a semidiscrete finite volume element method for the nonlinear convection–diffusion problem is considered. Under some regular assumptions, we obtain the L 2 and H 1 norm error estimates of numerical solution. Furthermore, we investigate two-grid finite volume element method for the considered equations. Compared with the standard method, the two-grid method is of the same order as the standard method in the H 1-norm as long as the mesh sizes satisfy . However, the two-grid method involves much less work than the standard method. Finally, some numerical results are provided to verify the established theoretical analysis. [Copyright &y& Elsevier]

A fully discrete two-grid finite-volume method (FVM) for a nonlinear parabolic problem is studied in this paper. This method involves solving a nonlinear parabolic equation on coarse mesh space and a linearized parabolic equation on fine grid. Both L2 and H1 norm error estimates of the standard FVM for the nonlinear parabolic problem are derived. Compared with the standard FVM, the two-level method is of the same order as the one-level method in the H1-norm as long as the mesh sizes satisfy h=O(H3/2). However, the two-level method involves much less work than the standard method. Numerical results are provided to demonstrate the effectiveness of our algorithm. [ABSTRACT FROM AUTHOR]

Abstract: In this article, new one- and two-level Newton iterative mixed finite element methods (MFEM) for two-dimensional stationary conduction–convection problems are presented. In both of the methods, we decouple the conduction–convection problem into two systems: one is for the temperature and the other is for the velocity and pressure. Thus much computational time can be saved. In the new Newton MFEM, we first solve the equation for the temperature, then solve a Navier–Stokes problem by Newton iterative method. While in the two-level method, we solve the decoupled conduction–convection problem as same as in the one-level method using a coarse grid firstly, then seek a fine grid solution by solving a linearized problem on a fine grid. Stability analysis is performed and error estimates of the methods are derived, which show that our methods are stable and have a good precision. Numerical experiments are also given, which confirm the theoretical analysis and demonstrate the efficiency of the new methods. [ABSTRACT FROM AUTHOR]