Conservation of mass and momentum of the least-squares spectral collocation scheme for the Stokes problem

From the literature it is known that spectral least-squares schemes perform poorly with respect to mass conservation and compensate this lack by a superior conservation of momentum. This should be revised, since the here presented new least-squares spectral collocation scheme leads to an outstanding performance with respect to conservation of momentum and mass. The reasons can be found in using only a few elements, each with high polynomial degree, avoiding normal equations for solving the overdetermined linear systems of equations and by introducing the Clenshaw–Curtis quadrature rule for imposing the average pressure to be zero. Furthermore, we combined the transformation of Gordon and Hall (transfinite mapping) with our least-squares spectral collocation scheme to discretize the internal flow problems.     

Analysis of a two-dimensional grade-two fluid model with a tangential boundary condition

This article studies the solutions in H¹ of a two-dimensional grade-two fluid model with a non-homogeneous Dirichlet tangential boundary condition, on a Lipschitz-continuous domain. Existence is proven by splitting the problem into a generalized Stokes problem and a transport equation, without restricting the size of the data and the constant parameters of the fluid. A substantial part of the article is devoted to a sharp analysis of this transport equation, under weak regularity assumptions. By means of this analysis, it is established that each solution of the grade-two fluid model satisfies energy equalities and converges strongly to a solution of the Navier–Stokes equations when the normal stress modulus α tends to zero. When the domain is a polygon, it is shown that the regularity of the solution is related to that of a Stokes problem. Uniqueness is established in a convex polygon, with adequate restrictions on the size of the data and parameters.     

Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods

In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, Q ₁^rot and EQ ₁^rot. Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. This project is supported in part by the National Natural Science Foundation of China (10471103) and is subsidized by the National Basic Research Program of China under the grant 2005CB321701.     

Study of the mixed finite volume method for Stokes and Navier‐Stokes equations

We present finite volume schemes for Stokes and Navier‐Stokes equations. These schemes are based on the mixed finite volume introduced in (Droniou and Eymard, Numer Math 105 (2006), 35‐71), and can be applied to any type of grid (without “orthogonality” assumptions as for classical finite volume methods) and in any space dimension. We present numerical results on some irregular grids, and we prove, for both Stokes and Navier‐Stokes equations, the convergence of the scheme toward a solution of the continuous problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

An error estimate of the proper orthogonal decomposition in model reduction and data compression

A kind of the proper orthogonal decomposition (POD) is used for data compression of rugged surface and reduction of the Navier–Stokes equations. An error estimate of the POD in model reduction and data compression is discussed. The numerical examples show that the error between the POD approximate solution and reference solution is consistent with theoretical results, and also show that the proposed algorithm is feasible and efficient. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A defect correction method for the time‐dependent Navier‐Stokes equations

A method for solving the time dependent Navier‐Stokes equations, aiming at higher Reynolds' number, is presented. The direct numerical simulation of flows with high Reynolds' number is computationally expensive. The method presented is unconditionally stable, computationally cheap, and gives an accurate approximation to the quantities sought. In the defect step, the artificial viscosity parameter is added to the inverse Reynolds number as a stability factor, and the system is antidiffused in the correction step. Stability of the method is proven, and the error estimations for velocity and pressure are derived for the one‐ and two‐step defect‐correction methods. The spacial error is O(h) for the one‐step defect‐correction method, and O(h²) for the two‐step method, where h is the diameter of the mesh. The method is compared to an alternative approach, and both methods are applied to a singularly perturbed convection–diffusion problem. The numerical results are given, which demonstrate the advantage (stability, no oscillations) of the method presented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Estimates of the Green’s Functions for the Elasto-static Equations and Stokes Equations in a Three Dimensional Lipschitz Domain

In this paper, we study a Green’s functions G ^E , G ^S for an elasto-static equations and Stokes equations in a three-dimensional bounded Lipschitz domain Ω. We prove that there is a positive constant c > 0 depending on the Lipschitz constant such that for all . Furthermore, we show that there is a positive constant η ∈ (0,1) depending on the Lipschitz constant such that for all . The second author is partially supported by Korea Research Foundation Grant KRF C-00005.     

Stokes问题的各向异性平行四边形有限元逼近

1 引言 Stokes问题是标准的混合问题,速度与压力同时计算,关于该问题有限元求解的文章很多(见文献[1-5])但大多都是基于对区域的正则剖分或拟一致剖分,即要求网格剖分满足hk/pK≤C,(A)K∈Jh,其中C>0为一常数,hk,pK分别为单元K的直径及内切园直径,在实际应用问题中,由于边界层或区域的拐角处需考虑物质的各向异性特征,此时对空间区域Q的剖分不再满足正则性或拟一致条件,而需要用各向异性网格剖分,才能更贴切地描述其真实情形.     

Hodge分解在最小耗散原理中的应用

就流体力学中的Helmholtz最小耗散原理的几种变分推导方法进行综述,利用Hodge分解定理给出一个新的推导方法.     

On the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domains

We present a uniqueness theorem for time-periodic solutions to the Navier–Stokes equations in unbounded domains. Thus far, results on the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domain, roughly speaking, have only found that a small time-periodic L ⁿ -solution is unique within the class of solutions which have sufficiently small L ^∞(L ⁿ )-norm. In this paper, we show that a small time-periodic L ⁿ -solution is unique within the class of all time-periodic L ⁿ -solutions, which contains large solutions. We also consider the uniqueness of solutions in weak-L ⁿ space. The proof of the present uniqueness theorem is based on the method of dual equations.