A two-grid algorithm based on Newton iteration for the stream function form of the Navier-Stokes equations

Abstract

In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navier-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinear system on the coarse mesh space and two similar linear systems (with same stiffness matrix but different right-hand side) on the fine mesh space.The convergence analysis and error estimation of the algorithm are given for the case of conforming elements. Furthermore, the algorithm produces a numerical solution with the optimal asymptotic H 2 -error. Finally, we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the Navier-Stokes equations. §1 Introduction Convergence analysis and error estimation for the finite element method of the primitive variable formulation of the Navier-Stokes(NS) equations have been extensively developed in recent years, e.g. [9,14,15,22,23]. In some research, the stream function formulation is preferred and commonly used for solving the NS problem. The one attraction of stream function formation of the NS equations [14] is that the pressure term can be canceled in the weak form, and there is only one scalar unknown to be solved. The representative work in this field is the analysis of convergence for the standard weak formulation by Cayco and Nicolaides [5,6].Two-grid algorithm is a very promising approach for solving the primitive variable formulation (velocity-pressure formulation) of the NS equations [13,18,20]. Moreover, it can be applied to the stream function formulation of the NS equations [10-12, 21, 26]. The two-grid method is attractive when h H, where H and h are mesh sizes for the coarse grid and fine grid respectively. In the two-grid method, the main point is that the scaling h is how to match H ν for a positive number ν so that the two-grid algorithm produces an approximate solution with Received: 2010-09-25. MR Subject Classification: 65N30, 65N12, 65H99, 65G99.

Modified two-grid method for solving coupled Navier-Stokes/Darcy model based on Newton iteration

“…The reason for it is that we use more accurate initial value φ h in the Newton iteration. More generally, if the initial value φ H is used like as in [7], the same accuracy can be achieved by applying the twostep Newton iteration, which almost has the same computational cost like one-step Newtion iteration(see [25]). …”

A two-level finite element discretization of the streamfunction formulation of the stationary quasi-geostrophic equations of the ocean

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