定常非对称流动方程的G/L-S稳定化有限元方法
|
摘要
定常非对称流动问题有着很重要的物理力学背景，已经在一些文章中给过这方面的综述，并且有些文章对该问题进行了严格的数学分析，得到了解的存在唯一性结果。然而对该问题的数值研究却很少，本文的目的就是在于通过稳定化有限元方法讨论其数值求解。本文之所以采用Galerkin/最小平方法有如下原因：(1)G/L-S稳定化有限元方法，它不要求BB条件成立，该方法应用到流体等线性问题上已经取得了很大成功，而对非线性问题却研究很少；(2)文章[1]是Galerkin/最小平方法[6]在N-S方程及[8]在非线性方程上地应用，已经取得了一定成功；(3)与文章[1]中N-S方程组相比而言，定常非对称流动方程组增多了方程(1.3)，因此理论和数值分析难度进一步增大；(4)本文从有限元角度分析，导出了问题的混合变分格式，证明了定常非对称流动方程变分格式解的存在唯一性。并建立了方程的G/L-S有限元离散格式，相应地证明了有限元离散解的存在唯一性。证明了有限元离散解对任意有限元空间的组合是稳定的。并给出有限元解的收敛性和误差估计。
The stationary dissymmetry flow problem has very important physical dynamics background that summarized in some papers. Some papers have processed severe mathematics analysis, gained existence and uniqueness of the solution. But few papers have numerical research of the problem. The present paper deals with the numerical solution by means of Stabilized Finite Element Methods. That G/L-S finite element discrete form is used in this paper has the following reasons: (1) a G/L-S stabilized finite element method doesn't request BB condition come into existence; (2)Paper[l] is an successful application of the G/L-S method [5] to N-S equations and its altrenative [6] to nonlinear equations; (3)Compared with N-S equations, the stationary dissymmetry flow equations are more complicated which increases the difficulties of theory and numerical analysis; (4)The mix variational form is educed and existence and uniqueness of the solution are proved. A G/L-S finite element discrete form is proposed. The discrete solution is stable for any combination of finite element spaces. But also the convergence and error estimate of the finite element solution are showed.
The stationary dissymmetry flow problem has very important physical dynamics background that summarized in some papers. Some papers have processed severe mathematics analysis, gained existence and uniqueness of the solution. But few papers have numerical research of the problem. The present paper deals with the numerical solution by means of Stabilized Finite Element Methods. That G/L-S finite element discrete form is used in this paper has the following reasons: (1) a G/L-S stabilized finite element method doesn't request BB condition come into existence; (2)Paper[l] is an successful application of the G/L-S method [5] to N-S equations and its altrenative [6] to nonlinear equations; (3)Compared with N-S equations, the stationary dissymmetry flow equations are more complicated which increases the difficulties of theory and numerical analysis; (4)The mix variational form is educed and existence and uniqueness of the solution are proved. A G/L-S finite element discrete form is proposed. The discrete solution is stable for any combination of finite element spaces. But also the convergence and error estimate of the finite element solution are showed.
引文
[1] Tian-Xiao Zhou, and Min-Fu Feng, A least squares Petrov-Galerldn finite element method for the stationary Navier-Stokes equations, Math. Comput. vol. 60. hum 202,April(1993),531-543.
[2] Petrosyan, L.G., Some problems of mechanics of fluids with antisymmetric stress tensor, Erevan,1984(in Russian)
[3] Lukaszewicz, G., On stationary flows of asymmetric fluids, Rend. Accad. Naz. Memorie di mathematica,Ⅺ (1987).
[4] Lukaszewicz, G., On stationary flows of asymmetric fluids with heat convection, Math. Meth. in Appl. Sci. 11, (1989),343-351.
[5] C.Johnson and J.Saranen, Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations, Math.Comp. 47(1986),1-18.
[6] T.J.R.Hughes, L.P.Franca, and G.M.Hulbert, A new finite element formulation for computational fluid dynamics, Ⅷ. The Galerkin/least-squares method for advectivediffusive equations, Comput.Methods Appl.Mech.Engrg. 89 (1989), 173-189.
[7] L.P.Franca and T.J.Hughes, Tow classes of mixed finite element methods, Comput.Methods Appl.Mech.Engrg. 69(1989),89-129.
[8] L.P.Franca,S.L.Frey and T.J.R.Hughes, Stabilized finite element methods.Ⅰ.Application to the advective-diffusion model, Preprint, Sept,1990.
[9] Alessandro Russo, Bubble stabilization of finite element methods for the linearized in compressible Navier-Stokes equations, Comput.Methods Appl.Mech.Engrg. 132(1996),335-343.
[10] V.Girault and P.A.Raviart, Finite element methods for Navier-Stokes equations, Lecture Notes in Math., vol.749, Spring-Verlag, Berlin and New York, 1981.
[11] P.G.Ciarlet, The finite element method for elliptic problems, North-Hollang, Amsterdam, 1978.
[12] L.Tobiska and G.Lube, A modified streamline diffusion method for solving the statinary N-S equations,Preprint.
[2] Petrosyan, L.G., Some problems of mechanics of fluids with antisymmetric stress tensor, Erevan,1984(in Russian)
[3] Lukaszewicz, G., On stationary flows of asymmetric fluids, Rend. Accad. Naz. Memorie di mathematica,Ⅺ (1987).
[4] Lukaszewicz, G., On stationary flows of asymmetric fluids with heat convection, Math. Meth. in Appl. Sci. 11, (1989),343-351.
[5] C.Johnson and J.Saranen, Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations, Math.Comp. 47(1986),1-18.
[6] T.J.R.Hughes, L.P.Franca, and G.M.Hulbert, A new finite element formulation for computational fluid dynamics, Ⅷ. The Galerkin/least-squares method for advectivediffusive equations, Comput.Methods Appl.Mech.Engrg. 89 (1989), 173-189.
[7] L.P.Franca and T.J.Hughes, Tow classes of mixed finite element methods, Comput.Methods Appl.Mech.Engrg. 69(1989),89-129.
[8] L.P.Franca,S.L.Frey and T.J.R.Hughes, Stabilized finite element methods.Ⅰ.Application to the advective-diffusion model, Preprint, Sept,1990.
[9] Alessandro Russo, Bubble stabilization of finite element methods for the linearized in compressible Navier-Stokes equations, Comput.Methods Appl.Mech.Engrg. 132(1996),335-343.
[10] V.Girault and P.A.Raviart, Finite element methods for Navier-Stokes equations, Lecture Notes in Math., vol.749, Spring-Verlag, Berlin and New York, 1981.
[11] P.G.Ciarlet, The finite element method for elliptic problems, North-Hollang, Amsterdam, 1978.
[12] L.Tobiska and G.Lube, A modified streamline diffusion method for solving the statinary N-S equations,Preprint.