[1] Babuška I, Osborn J. Eigenvalue problems//Handbook of numerical analysis. Vol Ⅱ. Handb Numer Anal, Ⅱ. Amsterdam:North-Holland, 1991:641-787
[2] Mercier B, Osborn J, Rappaz J, Raviart P A. Eigenvalue approximation by mixed and hybrid methods. Math Comp, 1981, 36:427-453
[3] Babuka I, Osborn J. Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math Comp, 1989, 52:275-297
[4] Lin Q, Xie H H. Asymptotic error expansion and Richardson extrapolation of eigenvalue approximations for second order elliptic problems by mixed finite element method. Appl Numer Math, 2009, 59:1184-1893
[5] Lin Q. Fourth order eigenvalue approximation by extrapolation on domains with reentrant corners. Numer Math, 1991, 58:631-640
[6] Jia S H, Xie H H, Gao S Q. Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods. Appl Math, 2009, 54:1-15
[7] Chen H T, Jia S H, Xie H H. Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem. Appl Numer Math, 2011, 61:615-629
[8] Chen H T, Jia S H, Xie H H. Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems. Appl Math, 2009, 54:237-250
[9] Lovadina C, Lyly M, Stenberg R. A posteriori estimates for the Stokes eigenvalue problem. Numer Meth Part Differ Equ, 2009, 25:244-257
[10] Jing F F, Han W M, Zhang Y C, Yan W J. Analysis of an a posteriori error estimator for a variational inequality governed by the Stokes equations. J Comput Appl Math, 2020, 372:112721
[11] Cliffe K A, Hall E, Houston P. Adaptive discontinuous Galerkin methods for eigenvalue problems arising in incompressible fluid flows. SIAM J Sci Comput, 2010, 31:4607-4632
[12] Yin X B, Xie H H, Jia S H, Gao S Q. Asymptotic expansions and extrapolations of eigenvalues for the Stokes problem by mixed finite element methods. J Comput Appl Math, 2008, 215:127-141
[13] Chen W, Lin Q. Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method. Appl Math, 2006, 51:73-88
[14] Huang P Z, He Y N, Feng X L. Numerical investigations on several stabilized finite element methods for the Stokes eigenvalue problem. Math Probl Eng, 2011, 2011:1-14
[15] Xu J C, Zhou A H. A two-grid discretization scheme for eigenvalue problems. Math Comp, 1999, 70:17-25
[16] Huang P Z, He Y N, Feng X L. Two-level stabilized finite element method for Stokes eigenvalue problem. Appl Math Mech Engl Ed, 2012, 33:62-630
[17] Armentano M G, Moreno V. A posteriori error estimates of stabilized low-order mixed finite elements for the generalized Stokes eigenvalue problem. J Comput Appl Math, 2014, 269:132-149
[18] Brezzi F, Pitkäranta J. On the stabilization of finite element approximations of the Stokes problems//Notes on Numerical Fluid Mechanics. Braunschweig:Vieweg, 1984
[19] Hughes J, Franca L, Balesra M. A new finite element formulation for computational fluid dynamics:V. Circunventing the Babuska-Brezzi condition:a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput Methods Appl Mech Engrg, 1986, 59:85-99
[20] Brezzi F, Douglas J. Stabilized mixed methods for the Stokes problem. Numer Math, 2002, 20:653-677
[21] Kechar N, Silvester D. Analysis of a locally stabilized mixed finite element method for the Stokes problem. Math Comp, 1992, 58:1-10
[22] Jing F F, Li J, Chen Z X, Zhang Z H. Numerical analysis of a characteristic stabilized finite element method for the time dependent Navier Stokes equations with nonlinear slip boundary conditions. J Comput Appl Math, 2017, 320:43-60
[23] Tobiska L, Verfürth R. Analysis of a streamline diffudion finite element method for the Stokes and NavierStokes equations. SIAM J Numer Anal, 1996, 58:107-127
[24] Araya R, Barrenechea G R, Valentin F. Stabilized finite element method based on multiscale enrichment for the Stokes problem. SIAM J Numer Anal, 2006, 44:322-348
[25] He Y N, Li J. A stabilized finite element method based on local ploynomail pressure projection for the stationary Navier-Stokes equation. Appl Numer Math, 2008, 58:1503-1514
[26] Burman E, Hansbo P. A Unified Stabilized Method for Stokes' and Darcy's Equations//Tech Report 2002-15. Sweden Göteborg:Chalmers Finite Element Center, 2002
[27] Barrenechea G, Valentin F. An unusual stabilized finite element method for a feneralized Stokes problem. Numer Math, 2002, 20:653-677
[28] Baiocchi C, Breezi F, Franca L. Virtual bubbles and Galerkin-least-squares type methods. Comput Methods Appl Mech Engrg, 1993, 105:125-141
[29] Russo A. Bubble stabilization of finite element methods for the linearized incompressible Navier-Stokes equations. Comput Methods Appl Mech Engrg, 1996, 132:335-343
[30] Franca L, Russo A. Approximation of the Stokes problem by residual-free macro bubbles. East-West J Numer Math, 1996, 4:265-278
[31] Adams R A. Sobolev Spaces. New York:Acadamic Press, 1975
[32] Giraut V, Raviart P A. Finite Element Method for Navier-Stokes Equations. Berlin:Springer-Verlag, 1986
[33] Clément P. Apprximation by finite element functions using local regularization. RAIRO Anal Numer, 1975, 9:77-84
[34] Quateroni A. Numerical Models for Differnetial Problems//MS&A Series, Vol 2. Milan:Springer-Verlag, 2009
[35] Ern A, Guermond J L. Theory and Practice of Finite Element. New York:Springer-Verlag, 2004
[36] Temam R. Navier-Stokes Equations, Theory and Numerical Analysis. Third Ed. Amsterdam:NorthHolland, 1984
[37] Ge Z H, Yan J J. Analysis of multiscale finite element method for the stationary Navier-Stokes equations. Nonlinear Anal RWA, 2012, 13:385-394
[38] Freefem++, version 3.13.3, < http://www.freefem.org/> |