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
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
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
Two basic hypothesises of Taylor-Galerkin Finite Element Method are studied in this paper. One of them which is unreasonable is redefined. The only hypothesis becomes the standpoint of Generalized Finite Element. We u...Two basic hypothesises of Taylor-Galerkin Finite Element Method are studied in this paper. One of them which is unreasonable is redefined. The only hypothesis becomes the standpoint of Generalized Finite Element. We use this idea to analysis stream function-vorticity equations with Modified Taylor-Galerkin Finite Element Method, and give the two-step solving method, which makes the solving process more reasonable than ever before. Several computational examples reveal that the results of this new method are satisfied.展开更多
For some special flows, especially the potential flow in a plane, using the hodograph method has obvious advantages. Realistic flows have a stream surface, namely, a two-dimensional manifold, on which the velocity vec...For some special flows, especially the potential flow in a plane, using the hodograph method has obvious advantages. Realistic flows have a stream surface, namely, a two-dimensional manifold, on which the velocity vector of the flow lies on its tangent space. By introducing a stream function and a potential function, we establish the hodo- graph method for potential flows on a surface using the tensor analysis. For the derived hodograph equation, we obtain a characteristic equation and its characteristic roots, from which we can classify the type of the second-order hodograph equation. Moreover, we give some examples for special surfaces.展开更多
Fourth-order stream-function methods are proposed for the time dependent, incom- pressible Navier-Stokes and Boussinesq equations. Wide difference stencils are used instead of compact ones and the boundary terms are h...Fourth-order stream-function methods are proposed for the time dependent, incom- pressible Navier-Stokes and Boussinesq equations. Wide difference stencils are used instead of compact ones and the boundary terms are handled by extrapolating the stream-function values inside the computational domain to grid points outside, up to fourth-order in the noslip condition. Formal error analysis is done for a simple model problem, showing that this extrapolation introduces numerical boundary layers at fifth-order in the stream-function. The fourth-order convergence in velocity of the proposed method for the full problem is shown numerically.展开更多
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Two basic hypothesises of Taylor-Galerkin Finite Element Method are studied in this paper. One of them which is unreasonable is redefined. The only hypothesis becomes the standpoint of Generalized Finite Element. We use this idea to analysis stream function-vorticity equations with Modified Taylor-Galerkin Finite Element Method, and give the two-step solving method, which makes the solving process more reasonable than ever before. Several computational examples reveal that the results of this new method are satisfied.
基金Project supported by the National Natural Science Foundation of China (Nos. 10971165, 10771167,and 10926080)
文摘For some special flows, especially the potential flow in a plane, using the hodograph method has obvious advantages. Realistic flows have a stream surface, namely, a two-dimensional manifold, on which the velocity vector of the flow lies on its tangent space. By introducing a stream function and a potential function, we establish the hodo- graph method for potential flows on a surface using the tensor analysis. For the derived hodograph equation, we obtain a characteristic equation and its characteristic roots, from which we can classify the type of the second-order hodograph equation. Moreover, we give some examples for special surfaces.
基金funding from NSF under grants DMS-0713670 and ACI-0204932funding from NSERC Canada that supported this work
文摘Fourth-order stream-function methods are proposed for the time dependent, incom- pressible Navier-Stokes and Boussinesq equations. Wide difference stencils are used instead of compact ones and the boundary terms are handled by extrapolating the stream-function values inside the computational domain to grid points outside, up to fourth-order in the noslip condition. Formal error analysis is done for a simple model problem, showing that this extrapolation introduces numerical boundary layers at fifth-order in the stream-function. The fourth-order convergence in velocity of the proposed method for the full problem is shown numerically.
