NONLINEAR GALERKIN METHOD FOR THE EXTERIOR NONSTATIONARY NAVIER-STOKES EQUATIONS

Abstract

Abstract: A new algorithm combining nonlinear Galerkin method and coupling method of finite element and boundary element is introduced to solve the exterior nonstationary Navier-Stokes equations The regularity of the coupling variational formulation and the convergence of the approximate solution corresponding to the algorithm are proved If the fine mesh h is choosed as coarse mesh H-sgure, the nonlinear Galerkin method, nonlinearity is only treated on the coarse grid and linearity is treated on the fine grid Hence, the new algorithm can save a large amount of computational time.

Key words: Navier-Stokes equations, finite element, boundary element, exterior, nonlinear, Galerkin method

2010 MSC Number:

HE Yin-nian;LI Kai-tai. NONLINEAR GALERKIN METHOD FOR THE EXTERIOR NONSTATIONARY NAVIER-STOKES EQUATIONS. Applied Mathematics and Mechanics (English Edition), 2002, 23(11): 1282-1291.

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References

[1] Marion M, Temam R. Nonlinear Galerkin methods:the finite elements case[J]. Numer Math, 1990,57(3):205-226.
[2] Marion M, XU Jin-chao. Error estimates on a new nonlinear Galerkin method based on two-grid finite elements[J]. SIAM J Numer Anal, 1995,32(4): 1170-1184.
[3] Ait Ou Ammi A, Marion M. Nonlinear Galerkin nethods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations[J]. Numer Math, 1994,68(2): 189-213.
[4] LI Kai-tai, ZHOU Lei. Finite element nonlinear Galerkin mothod for the penalty Navier-Stokes equations[J]. Numerical Mathematics Sinica. 1995,17(4): 360-380. (in Chinese)
[5] Johnson C, Nedelec J C. On the coupling of boundary integral and finite element methods[J]. Math Comp, 1980,35(152): 1063-1079.
[6] ZHU Jia-lin. Boundary Element Analysis for Elliptic Boundary Value Problem[M]. Beijing: Science Press, 1991. (in Chinese)
[7] Yu D H. The coupling of natural BEM and FEM for the Stokes problem on unbounded domain[J]. Chinese J Numer Math Appl, 1992,14 (4): 111-120.
[8] Sequeira A. The coupling of boundary integral and finite element methods for the bidimensional exterior steady Stokes problem[J]. Math Methvds ApplSci, 1983,5(3):356-375
[9] HE Yin-nian , LI Kai-tai. The coupling of boundary element and finite element methods for exterior nonstationary Stokes problem[J]. Acta Mathematica Scientia, 1991,11(2): 190-207.
[10] LI Kai-tai , HE Yin-nian. Stokes coupling method for the exterior flow, Part Ⅱ: Well-posedness analysis[J]. Acta Mathematica Scientia, 1996,16 (4): 442-457.
[11] HE Yin-nian, LI Kai-tai. The coupling of boundary integral and finite element methods for the nonstationary exterior flow[J]. Numer Methods for Partial Differential Equations, 1988,14 (5): 549-565.
[12] Girault V, Raviart P A. Finite Element Methods for Navier-Stokes Equations[M]. Heidelberg: Springer, 1986.
[13] Bernardi C, Raugel G A. Conforming finite element method for the time-dependent Navier-Stokes equations[J]. SIAM J Numer Anal, 1985,22(3):455-473.
[14] Temam R. Navier-Stokes Equations[M]. Amsterdam:North-Holland,1984.
[15] Ladyzhenskava Q A. The Mathematical Theory of Viscous Incompressible Flow[M]. New York: Gordon and Breach, 1969.
[16] Lions J L, Magenes E. Non-Homogeneous Boundary Value Problem and Applications[M]. Berlin, New York: Springer-Verlag, 1972.