In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinea...In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-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 Mgorithm 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.展开更多
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...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.展开更多
The finite element solution of a generalized Stokes system in terms of the flow variables stream function and vorticity is studied. This system results from a time discretization of the time-dependent Stokes system in...The finite element solution of a generalized Stokes system in terms of the flow variables stream function and vorticity is studied. This system results from a time discretization of the time-dependent Stokes system in stream function-vorticity formulation, or yet by the application of the characteristics method to solve the Navier-Stokes equations in the same representation. Numerical results presented for both cases illustrate the good behaviour of the adopted approach.展开更多
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 met...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.展开更多
A nonlinear Galerkin finite element method is presented for the two dimensional incom- pressible Navier-Stokes equations with stream-vorticity form.the scheme is based on two finite ele- ment spaces XH and XH for the ...A nonlinear Galerkin finite element method is presented for the two dimensional incom- pressible Navier-Stokes equations with stream-vorticity form.the scheme is based on two finite ele- ment spaces XH and XH for the approximation of the stream and vorticity function ,defined respec- tively on a coarse grid with grid size H and a fine grid with grid size h<<H.We prove that the difference between the new nonlinear Galerkin method and the standard Galerkin method is of the order H2both in stream function and vorticity.展开更多
A general theorem for the Stokes flow over a plane boundary with mixed stick-slip boundary conditions is established. This is done by using a representation for the velocity and pressure fields in the three-dimensiona...A general theorem for the Stokes flow over a plane boundary with mixed stick-slip boundary conditions is established. This is done by using a representation for the velocity and pressure fields in the three-dimensional Stokes flow in terms of a biharmonic function and a harmonic function. The earlier theorem for the Stokes flow due to fundamental singularities before a no-slip plane boundary is shown to be a special case of the present theorem. Furthermore, in terms of the Stokes stream function, a corollary of the theorem is also derived, providing a solution to the problem of the axisymmetric Stokes flow along a rigid plane with stick-slip boundary conditions. The formulae for the drag and torque exerted by the fluid on the boundary are established. An illustrative example is given.展开更多
In this paper, problems of the flow over a fat plate in the large Reynolds numbercase are studied by using the method of multiple scales￣[1,2].We have obtained N-orderuniformly valid asymptotic solutions of the Naver...In this paper, problems of the flow over a fat plate in the large Reynolds numbercase are studied by using the method of multiple scales￣[1,2].We have obtained N-orderuniformly valid asymptotic solutions of the Naver-Stodes equations.展开更多
基金supported by National Foundation of Natural Science under the Grant 11071216
文摘In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-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 Mgorithm 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.
文摘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.
文摘The finite element solution of a generalized Stokes system in terms of the flow variables stream function and vorticity is studied. This system results from a time discretization of the time-dependent Stokes system in stream function-vorticity formulation, or yet by the application of the characteristics method to solve the Navier-Stokes equations in the same representation. Numerical results presented for both cases illustrate the good behaviour of the adopted approach.
文摘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.
文摘A nonlinear Galerkin finite element method is presented for the two dimensional incom- pressible Navier-Stokes equations with stream-vorticity form.the scheme is based on two finite ele- ment spaces XH and XH for the approximation of the stream and vorticity function ,defined respec- tively on a coarse grid with grid size H and a fine grid with grid size h<<H.We prove that the difference between the new nonlinear Galerkin method and the standard Galerkin method is of the order H2both in stream function and vorticity.
文摘A general theorem for the Stokes flow over a plane boundary with mixed stick-slip boundary conditions is established. This is done by using a representation for the velocity and pressure fields in the three-dimensional Stokes flow in terms of a biharmonic function and a harmonic function. The earlier theorem for the Stokes flow due to fundamental singularities before a no-slip plane boundary is shown to be a special case of the present theorem. Furthermore, in terms of the Stokes stream function, a corollary of the theorem is also derived, providing a solution to the problem of the axisymmetric Stokes flow along a rigid plane with stick-slip boundary conditions. The formulae for the drag and torque exerted by the fluid on the boundary are established. An illustrative example is given.
文摘In this paper, problems of the flow over a fat plate in the large Reynolds numbercase are studied by using the method of multiple scales￣[1,2].We have obtained N-orderuniformly valid asymptotic solutions of the Naver-Stodes equations.
