In this paper, we first propose a new stabilized finite element method for the Stokes eigenvalue problem. This new method is based on multiscale enrichment, and is derived from the Stokes eigenvalue problem itself. The convergence of this new stabilized method is proved and the optimal priori error estimates for the eigenfunctions and eigenvalues are also obtained. Moreover, we combine this new stabilized finite element method with the two-level method to give a new two-level stabilized finite element method for the Stokes eigenvalue problem. Furthermore, we have proved a priori error estimates for this new two-level stabilized method. Finally, numerical examples confirm our theoretical analysis and validate the high effectiveness of the new methods.
Juan WEN
,
Pengzhan HUANG
,
Ya-Ling HE
. THE TWO-LEVEL STABILIZED FINITE ELEMENT METHOD BASED ON MULTISCALE ENRICHMENT FOR THE STOKES EIGENVALUE PROBLEM[J]. Acta mathematica scientia, Series B, 2021
, 41(2)
: 381
-396
.
DOI: 10.1007/s10473-021-0204-3
[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/>
