Abstract
In this paper, we propose a two-level nonconforming rotated finite element (TNRFE) method for solving the Navier–Stokes equations. A new nonconforming rotated finite element (NRFE) method was proposed by Douglas added by conforming bubbles to velocity and discontinuous piecewise constant to the pressure on quadrilateral elements possessing favorable stability properties. The TNRFE method involves solving a small Navier–Stokes problem on a coarse mesh with mesh size H and a large linearized Navier–Stokes problem on a fine mesh with mesh size h by the NRFE method. If we choose h = O H 2 , the TNRFE method gives the convergence rate of the same order as that of the NRFE method. Compared with the NRFE method, the TNRFE method can save a large amount of CPU time. In this paper, the stability of the approximate solutions and the error estimates are proved. Finally, the numerical experiments are given, and results indicate that the method is practicable and effective.
Highlights
When the Navier–Stokes equations are discretized by finite element methods, two problems often arise: one is that the discrete inf-sup condition is broken, and the other is that the pseudo-oscillation is caused by the dominant convection term
In order to solve the first problem, we use the new nonconforming element proposed by Jim Douglas [1] added by conforming bubbles to the velocity and discontinuous piecewise constant to the pressure on quadrilateral elements which possess favorable stability properties
Many methods have been proposed, including the two-level method. e aim of the twolevel method is to obtain the approximate solution of the nonlinear equation in less time and to maintain the optimal convergence speed. e concrete calculation is to solve a nonlinear problem on the coarse grid and a linear problem on the fine grid. e theoretical and numerical experiments in this paper show that, compared with the nonconforming rotated finite element (NRFE) method, the two-level nonconforming rotated finite element (TNRFE) method can save a lot of time when the convergence rate reaches the same order, which shows that it is an effective algorithm
Summary
When the Navier–Stokes equations are discretized by finite element methods, two problems often arise: one is that the discrete inf-sup condition is broken, and the other is that the pseudo-oscillation is caused by the dominant convection term. Mathematical Problems in Engineering linear triangle element with three midpoints as the velocity approximation space and the piecewise constant finite element as the pressure approximation space to obtain an approximation scheme (the C-R scheme) for the Stokes problem In this way, the LBB condition is satisfied and some optimal error estimates of velocity and pressure are obtained. If the definition of the global nonconforming space which requires continuity at the midpoint of the common inner boundary of the adjacent elements is adopted, the optimal error estimates for the real quadrilateral subdivision region will not be obtained To solve this problem, Jim Douglas et al [1] modified the rotated bilinear local basis span 1, x, y, x2 − y2 to span 1, x, y, (3x2 − 5x4) − (3y2 − 5y4)}, which has the following properties: < 1, wj − wk > Γjk 0 and Γj 0.
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