The two-grid finite element approximation to the stream function form of the stationary Navier-Stokes equations was analyzed. The algorithms involve solving one small, nonlinear coarse mesh system, one linear problem on the fine mesh system, and a linear correct problem on the coarse mesh. The algorithms with the correct problem and without the correct problem were discussed. The algorithms produce an approximate solution with the optimal, asymptotic accuracy for any fixed Reynolds number.
Residual-based a posteriori error estimate for conforming finite element solutions of incompressible Navier-Stokes equations, which is computed with a new two-level method that is different from Volker John, is derived. A posteriori error estimate contains additional terms in comparison to the estimate for the solution obtained by the standard finite element method. The importance of the additional terms in the error estimates is investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than the convergence of discrete solution. The two-level method aims to solve the nonlinear problem on a coarse grid with less computational work, then to solve the linear problem on a fine grid, which is superior to the usual finite element method solving a similar nonlinear problem on the fine grid.Key words Finite element method, Navier-Stokes equations, residual-based a posteriori error estimate, two-level method.
A two-grid technique for solving the steady incompressible Navier-Stokes equations in a penalty method was presented and 1 1 the convergence of numerical solutions was analyzed. If a coarse size H and a fine size h satisfy H = O(h~-*)(s = 0( n = 2) ;s = ~-( n : 3) ,where n is a space dimension), this method has the same convergence accuracy as the usual finite element method. But the two-grid method can save a lot of computation time for its brief calculation. Moreover, a numerical test was couducted in order to verify the correctness of above theoretical analysis.
Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.
Two-level finite element approximation to stream function form of unsteady Navier-Stokes equations is studied. This algorithm involves solving one nonlinear system on a coarse grid and one linear problem on a fine grid. Moreover, the scaling between these two grid sizes is super-linear. Approximation, stability and convergence aspects of a fully discrete scheme are analyzed. At last a numrical example is given whose results show that the algorithm proposed in this paper is effcient. § 1 Introduction Numerical solutions of nonlinear systems arising in discretization of incompressibleNavier-Stokes equations may be very time-consuming and some difficulties appear inevitably because of incompressibility of fluids and nonlinear terms in the equations. For these reasons ,we will study two-level method for the stream function form of the unsteady Navier-Stokes equations in this paper. Advantages of the stream function form are that the incompressibility condition will be satisfied automatically and pressure will not be presented in weak form. Main purpose of the two-level method is to compute discrete approximation of solutions of nonlinear partial differential equations with less computation work and to preserve optimal orders of convergence at the same time. The basis of the twolevel method is a coarse grid and a fine grid. At first,the given problems will be solved on the coarse grid, which is in general inexpensive. The second step is to do some numerical work in doing one or a few steps of iteration for solving a linear problem on the fine grid.Compared to usual finite element method which solves the given nonlinear problems on the fine grid,two-level method can save a lot of computation time with the same convergence accuracy.Two-level method has gained some attraction in the last couple of years. In references
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