# American Institute of Mathematical Sciences

August  2013, 7(3): 795-811. doi: 10.3934/ipi.2013.7.795

## Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods

 1 LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 2 LSEC, NCMIS, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Received  June 2012 Revised  December 2012 Published  September 2013

A short survey of lower bounds of eigenvalue problems by nonconforming finite element methods is given. The class of eigenvalue problems considered covers Laplace, Steklov, biharmonic and Stokes eigenvalue problems.
Citation: Qun Lin, Hehu Xie. Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Problems & Imaging, 2013, 7 (3) : 795-811. doi: 10.3934/ipi.2013.7.795
##### References:
 [1] H. Ahn, Vibration of a pendulum consisiting of a bob suspended from a wire,, Quart. Appl. Math., 39 (1981), 109.   Google Scholar [2] A. Andreev and T. Todorov, Isoparametric finite-element approximation of a Steklov eigenvalue problem,, IMA J. Numer. Anal., 24 (2004), 309.  doi: 10.1093/imanum/24.2.309.  Google Scholar [3] M. Armentano and R. Durán, Asymptotic lower bounds for eigenvalues by nonconforming finite element methods,, Electron. Trans. Numer. Anal., 17 (2004), 93.   Google Scholar [4] I. Babuška and J. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems,, Math. Comp., 52 (1989), 275.  doi: 10.2307/2008468.  Google Scholar [5] I. Babuška and J. Osborn, Eigenvalue Problems, in handbook of numerical analysis,, Vol. II, (1991), 641.   Google Scholar [6] C. Bacuta and J. Bramble, Regularity estimates for the solutions of the equations of linear elasticity in convex plane polygonal domain,, Special issue dedicated to Lawrence E. Payne, 54 (2003), 874.   Google Scholar [7] C. Bacuta, J. Bramble and J. Pasciak, "Shift Theorems for the Biharmonic Dirichlet Problem,", Recent Progress in Computational and Appl. PDEs, (2001).   Google Scholar [8] P. Batcho and G. Karniadakis, Generalized Stokes eigenfunctions: A new trial basis for the solution of the incompressible Navier-Stokes equations,, J. Comput. Phys., 115 (1994), 121.  doi: 10.1006/jcph.1994.1182.  Google Scholar [9] S. Bergman and M. Schiffer, "Kernel Functions and Elliptic Differential Equations in Mathematical Physics,", Academic Press, (1953).   Google Scholar [10] A. Bermudez, R. Rodriguez and D. Santamarina, A finite element solution of an added mass formulation for coupled fluid-solid vibrations,, Numer. Math., 87 (2000), 201.  doi: 10.1007/s002110000175.  Google Scholar [11] J. Bramble and J. Osborn, Approximation of Steklov eigenvalues of non-selfadjoint second order elliptic operators,, in, (1972), 387.   Google Scholar [12] F. Brezzi and M. Fortin, "Mixed and Hybrid Finite Element Methods,", Springer-Verlag, (1991).  doi: 10.1007/978-1-4612-3172-1.  Google Scholar [13] D. Bucur and I. Ionescu, Asymptotic analysis and scaling of friction parameters,, Z. Angew. Math. Phys. (ZAMP), 57 (2006), 1042.  doi: 10.1007/s00033-006-0070-9.  Google Scholar [14] S. Brenner and L. Scott, "The Mathematical Theory of Finite Element Methods,", Texts in Applied Mathematics, (1994).   Google Scholar [15] F. Chatelin, "Spectral Approximation of Linear Operators,", With solutions to exercises by Mario Ahués. Computer Science and Applied Mathematics, (1983).   Google Scholar [16] P. Ciarlet, "The Finite Element Method for Elliptic Problem,", Studies in Mathematics and its Applications, (1978).   Google Scholar [17] C. Conca, J. Planchard and M. Vanninathanm, "Fluid and Periodic Structures,", John Wiley & Sons, (1995).   Google Scholar [18] K. Feng, A difference scheme based on variational principle,, Appl. Math and Comp. Math, 2 (1965), 238.   Google Scholar [19] V. Girault and P. Raviart, "Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms,", Springer Series in Computational Mathematics, (1986).  doi: 10.1007/978-3-642-61623-5.  Google Scholar [20] P. Grisvard, "Singularities in Boundary Problems," MASSON and Springer-Verlag,, 1985., ().   Google Scholar [21] D. Hinton and J. Shaw, Differential operators with spectral parameter incompletely in the boundary conditions,, Funkcialaj Ekvacioj (Serio Internacial), 33 (1990), 363.   Google Scholar [22] J. Hu, Y. Q. Huang and H. Shen, The lower approximation of eigenvalue by lumped mass finite element methods,, J. Comput. Math., 22 (2004), 545.   Google Scholar [23] J. Hu, Y. Huang and Q. Lin, The lower bounds for eigenvalues of elliptic operators-by nonconforming finite element methods,, Submitted on Dec. 6 2011, (2011).  doi: 10.1007/s10915-013-9744-6.  Google Scholar [24] E. Leriche and G. Labrosse, Stokes eigenmodes in square domain and the stream function-vorticity correlation,, J. Comput. Phys., 200 (2004), 489.  doi: 10.1016/j.jcp.2004.03.017.  Google Scholar [25] Y. Li, Lower approximation of eigenvalue by the nonconforming finite element method,, (Chinese) Math. Numer. Sin., 30 (2008), 195.   Google Scholar [26] Y. Li, The lower bounds of eigenvalues by the Wilson element in any dimension,, Adv. Appl. Math. Mech., 3 (2011), 598.   Google Scholar [27] Q. Li, Q. Lin and H. Xie, Nonconforming finite element approximations of the steklov eigenvalue problem and its lower bound approximations,, Appl. Math., 58 (2013), 129.  doi: 10.1007/s10492-013-0007-5.  Google Scholar [28] Q. Lin, H. Huang and Z. Li, New expansions of numerical eigenvalues for $- \Delta u=\lambda\rho u$ by nonconforming elements,, Math. Comput., 77 (2008), 2061.  doi: 10.1090/S0025-5718-08-02098-X.  Google Scholar [29] Q. Lin and J. Lin, "Finite Element Methods: Accuracy and Improvement,", Science Press: Beijing, (2006).   Google Scholar [30] Q. Lin, L. Tobiska and A. Zhou, Superconvergence and extrapolation of nonconforming low order finite elements applied to the Poisson equation,, IMA. J. Numer. Anal., 25 (2005), 160.  doi: 10.1093/imanum/drh008.  Google Scholar [31] Q. Lin and H. Xie, The asymptotic lower bounds of eigenvalue problems by nonconforming finite element methods,, (Chinese) Mathematics in Practice and Theory, 42 (2012), 219.   Google Scholar [32] Q. Lin, H. Xie, F. Luo, Y. Li and Y. Yang, Stokes eigenvalue approximation from below with nonconforming mixed finite element methods,, (Chinese) Math. in Practice and Theory, 40 (2010), 157.   Google Scholar [33] Q. Lin, H. Xie and J. Xu, Lower bounds of the discretization for piecewise polynomials,, accepted by Math. Comp., (2011).   Google Scholar [34] H. Liu and L. Liu, Expansion and extrapolation of the eigenvalue on $Q_1^{rot}$ element,, Journal of Hebei University, 23 (2005), 11.   Google Scholar [35] H. Liu and N. Yan, Four finite element solutions and comparisions of problem for the Poisson equation eigenvalue,, J. Numer. Method. & Comput. Appl., 2 (2005), 81.   Google Scholar [36] F. Luo, Q. Lin and H. Xie, Computing the lower and upper bounds of Laplace eigenvalue problem: By combining conforming and nonconforming finite element methods,, Sci. China Math., 55 (2012), 1069.  doi: 10.1007/s11425-012-4382-2.  Google Scholar [37] B. Mercier, J. Osborn, J. Rappaz and P. Raviart, Eigenvalue approximation by mixed and hybrid methods,, Math. Comput., 36 (1981), 427.  doi: 10.1090/S0025-5718-1981-0606505-9.  Google Scholar [38] J. Osborn, Approximation of the eigenvalue of a nonselfadjoint operator arising in the study of the stability of stationary solutions of the Navier-Stokes equations,, SIAM J. Numer. Anal., 13 (1976), 185.  doi: 10.1137/0713019.  Google Scholar [39] R. Rannacher, Nonconforming finite element methods for eigenvalue problems in linear plate theory,, Numer. Math., 33 (1979), 23.  doi: 10.1007/BF01396493.  Google Scholar [40] G. Stang and G. Fix, "An Analysis of the Finite Element Method,", Prentice-Hall Series in Automatic Computation. Prentice-Hall, (1973).   Google Scholar [41] Y. Yang, "Finite Element Methods Analysis to Eigenvalue Problem,", Guizhou People Press: Guizhou, (2004).   Google Scholar [42] Y. Yang and H. Bi, Lower spectral bounds by Wilson's brick discretization,, Appl. Numer. Math., 60 (2010), 782.  doi: 10.1016/j.apnum.2010.03.019.  Google Scholar [43] Y. Yang and Z. Chen, The order-preserving convergence for spectral approximation of self-adjoint completely continuous operators,, Science in China Series A, 51 (2008), 1232.  doi: 10.1007/s11425-008-0002-6.  Google Scholar [44] Y. Yang, Q. Li and S. Li, Nonconforming finite element approximations of the steklov eigenvalue problem,, Appl. Numer. Math., 59 (2009), 2388.  doi: 10.1016/j.apnum.2009.04.005.  Google Scholar [45] Y. Yang, Q. Lin, H. Bi and Q. Li, Eigenvalue approximations from below using Morley elements,, Adv. Comput. Math., 36 (2012), 443.  doi: 10.1007/s10444-011-9185-4.  Google Scholar [46] Y. Yang, Z. Zhang and F. Lin, Eigenvalue approximation from below using nonforming finite elements,, Sci. China Math., 53 (2010), 137.  doi: 10.1007/s11425-009-0198-0.  Google Scholar [47] C. Yao and Z. Qiao, Extrapolation of mixed finite element approximations for the maxwell eigenvalue problem,, Numer. Math. Theory Methods Appl., 4 (2011), 379.  doi: 10.4208/nmtma.2011.m1018.  Google Scholar [48] X. Yin, H. Xie, S. Jiang and S. Gao, Asymptotic expansions and extrapolations of eigenvalues for the stokes problem by mixed finite element methods,, J. Comput. Appl. Math., 215 (2008), 127.  doi: 10.1016/j.cam.2007.03.028.  Google Scholar [49] Z. Zhang, Y. Yang and Z. Chen, Eigenvalue approximation from below by Wilson's element,, (Chinese) Math. Numer. Sin., 29 (2007), 319.   Google Scholar

show all references

##### References:
 [1] H. Ahn, Vibration of a pendulum consisiting of a bob suspended from a wire,, Quart. Appl. Math., 39 (1981), 109.   Google Scholar [2] A. Andreev and T. Todorov, Isoparametric finite-element approximation of a Steklov eigenvalue problem,, IMA J. Numer. Anal., 24 (2004), 309.  doi: 10.1093/imanum/24.2.309.  Google Scholar [3] M. Armentano and R. Durán, Asymptotic lower bounds for eigenvalues by nonconforming finite element methods,, Electron. Trans. Numer. Anal., 17 (2004), 93.   Google Scholar [4] I. Babuška and J. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems,, Math. Comp., 52 (1989), 275.  doi: 10.2307/2008468.  Google Scholar [5] I. Babuška and J. Osborn, Eigenvalue Problems, in handbook of numerical analysis,, Vol. II, (1991), 641.   Google Scholar [6] C. Bacuta and J. Bramble, Regularity estimates for the solutions of the equations of linear elasticity in convex plane polygonal domain,, Special issue dedicated to Lawrence E. Payne, 54 (2003), 874.   Google Scholar [7] C. Bacuta, J. Bramble and J. Pasciak, "Shift Theorems for the Biharmonic Dirichlet Problem,", Recent Progress in Computational and Appl. PDEs, (2001).   Google Scholar [8] P. Batcho and G. Karniadakis, Generalized Stokes eigenfunctions: A new trial basis for the solution of the incompressible Navier-Stokes equations,, J. Comput. Phys., 115 (1994), 121.  doi: 10.1006/jcph.1994.1182.  Google Scholar [9] S. Bergman and M. Schiffer, "Kernel Functions and Elliptic Differential Equations in Mathematical Physics,", Academic Press, (1953).   Google Scholar [10] A. Bermudez, R. Rodriguez and D. Santamarina, A finite element solution of an added mass formulation for coupled fluid-solid vibrations,, Numer. Math., 87 (2000), 201.  doi: 10.1007/s002110000175.  Google Scholar [11] J. Bramble and J. Osborn, Approximation of Steklov eigenvalues of non-selfadjoint second order elliptic operators,, in, (1972), 387.   Google Scholar [12] F. Brezzi and M. Fortin, "Mixed and Hybrid Finite Element Methods,", Springer-Verlag, (1991).  doi: 10.1007/978-1-4612-3172-1.  Google Scholar [13] D. Bucur and I. Ionescu, Asymptotic analysis and scaling of friction parameters,, Z. Angew. Math. Phys. (ZAMP), 57 (2006), 1042.  doi: 10.1007/s00033-006-0070-9.  Google Scholar [14] S. Brenner and L. Scott, "The Mathematical Theory of Finite Element Methods,", Texts in Applied Mathematics, (1994).   Google Scholar [15] F. Chatelin, "Spectral Approximation of Linear Operators,", With solutions to exercises by Mario Ahués. Computer Science and Applied Mathematics, (1983).   Google Scholar [16] P. Ciarlet, "The Finite Element Method for Elliptic Problem,", Studies in Mathematics and its Applications, (1978).   Google Scholar [17] C. Conca, J. Planchard and M. Vanninathanm, "Fluid and Periodic Structures,", John Wiley & Sons, (1995).   Google Scholar [18] K. Feng, A difference scheme based on variational principle,, Appl. Math and Comp. Math, 2 (1965), 238.   Google Scholar [19] V. Girault and P. Raviart, "Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms,", Springer Series in Computational Mathematics, (1986).  doi: 10.1007/978-3-642-61623-5.  Google Scholar [20] P. Grisvard, "Singularities in Boundary Problems," MASSON and Springer-Verlag,, 1985., ().   Google Scholar [21] D. Hinton and J. Shaw, Differential operators with spectral parameter incompletely in the boundary conditions,, Funkcialaj Ekvacioj (Serio Internacial), 33 (1990), 363.   Google Scholar [22] J. Hu, Y. Q. Huang and H. Shen, The lower approximation of eigenvalue by lumped mass finite element methods,, J. Comput. Math., 22 (2004), 545.   Google Scholar [23] J. Hu, Y. Huang and Q. Lin, The lower bounds for eigenvalues of elliptic operators-by nonconforming finite element methods,, Submitted on Dec. 6 2011, (2011).  doi: 10.1007/s10915-013-9744-6.  Google Scholar [24] E. Leriche and G. Labrosse, Stokes eigenmodes in square domain and the stream function-vorticity correlation,, J. Comput. Phys., 200 (2004), 489.  doi: 10.1016/j.jcp.2004.03.017.  Google Scholar [25] Y. Li, Lower approximation of eigenvalue by the nonconforming finite element method,, (Chinese) Math. Numer. Sin., 30 (2008), 195.   Google Scholar [26] Y. Li, The lower bounds of eigenvalues by the Wilson element in any dimension,, Adv. Appl. Math. Mech., 3 (2011), 598.   Google Scholar [27] Q. Li, Q. Lin and H. Xie, Nonconforming finite element approximations of the steklov eigenvalue problem and its lower bound approximations,, Appl. Math., 58 (2013), 129.  doi: 10.1007/s10492-013-0007-5.  Google Scholar [28] Q. Lin, H. Huang and Z. Li, New expansions of numerical eigenvalues for $- \Delta u=\lambda\rho u$ by nonconforming elements,, Math. Comput., 77 (2008), 2061.  doi: 10.1090/S0025-5718-08-02098-X.  Google Scholar [29] Q. Lin and J. Lin, "Finite Element Methods: Accuracy and Improvement,", Science Press: Beijing, (2006).   Google Scholar [30] Q. Lin, L. Tobiska and A. Zhou, Superconvergence and extrapolation of nonconforming low order finite elements applied to the Poisson equation,, IMA. J. Numer. Anal., 25 (2005), 160.  doi: 10.1093/imanum/drh008.  Google Scholar [31] Q. Lin and H. Xie, The asymptotic lower bounds of eigenvalue problems by nonconforming finite element methods,, (Chinese) Mathematics in Practice and Theory, 42 (2012), 219.   Google Scholar [32] Q. Lin, H. Xie, F. Luo, Y. Li and Y. Yang, Stokes eigenvalue approximation from below with nonconforming mixed finite element methods,, (Chinese) Math. in Practice and Theory, 40 (2010), 157.   Google Scholar [33] Q. Lin, H. Xie and J. Xu, Lower bounds of the discretization for piecewise polynomials,, accepted by Math. Comp., (2011).   Google Scholar [34] H. Liu and L. Liu, Expansion and extrapolation of the eigenvalue on $Q_1^{rot}$ element,, Journal of Hebei University, 23 (2005), 11.   Google Scholar [35] H. Liu and N. Yan, Four finite element solutions and comparisions of problem for the Poisson equation eigenvalue,, J. Numer. Method. & Comput. Appl., 2 (2005), 81.   Google Scholar [36] F. Luo, Q. Lin and H. Xie, Computing the lower and upper bounds of Laplace eigenvalue problem: By combining conforming and nonconforming finite element methods,, Sci. China Math., 55 (2012), 1069.  doi: 10.1007/s11425-012-4382-2.  Google Scholar [37] B. Mercier, J. Osborn, J. Rappaz and P. Raviart, Eigenvalue approximation by mixed and hybrid methods,, Math. Comput., 36 (1981), 427.  doi: 10.1090/S0025-5718-1981-0606505-9.  Google Scholar [38] J. Osborn, Approximation of the eigenvalue of a nonselfadjoint operator arising in the study of the stability of stationary solutions of the Navier-Stokes equations,, SIAM J. Numer. Anal., 13 (1976), 185.  doi: 10.1137/0713019.  Google Scholar [39] R. Rannacher, Nonconforming finite element methods for eigenvalue problems in linear plate theory,, Numer. Math., 33 (1979), 23.  doi: 10.1007/BF01396493.  Google Scholar [40] G. Stang and G. Fix, "An Analysis of the Finite Element Method,", Prentice-Hall Series in Automatic Computation. Prentice-Hall, (1973).   Google Scholar [41] Y. Yang, "Finite Element Methods Analysis to Eigenvalue Problem,", Guizhou People Press: Guizhou, (2004).   Google Scholar [42] Y. Yang and H. Bi, Lower spectral bounds by Wilson's brick discretization,, Appl. Numer. Math., 60 (2010), 782.  doi: 10.1016/j.apnum.2010.03.019.  Google Scholar [43] Y. Yang and Z. Chen, The order-preserving convergence for spectral approximation of self-adjoint completely continuous operators,, Science in China Series A, 51 (2008), 1232.  doi: 10.1007/s11425-008-0002-6.  Google Scholar [44] Y. Yang, Q. Li and S. Li, Nonconforming finite element approximations of the steklov eigenvalue problem,, Appl. Numer. Math., 59 (2009), 2388.  doi: 10.1016/j.apnum.2009.04.005.  Google Scholar [45] Y. Yang, Q. Lin, H. Bi and Q. Li, Eigenvalue approximations from below using Morley elements,, Adv. Comput. Math., 36 (2012), 443.  doi: 10.1007/s10444-011-9185-4.  Google Scholar [46] Y. Yang, Z. Zhang and F. Lin, Eigenvalue approximation from below using nonforming finite elements,, Sci. China Math., 53 (2010), 137.  doi: 10.1007/s11425-009-0198-0.  Google Scholar [47] C. Yao and Z. Qiao, Extrapolation of mixed finite element approximations for the maxwell eigenvalue problem,, Numer. Math. Theory Methods Appl., 4 (2011), 379.  doi: 10.4208/nmtma.2011.m1018.  Google Scholar [48] X. Yin, H. Xie, S. Jiang and S. Gao, Asymptotic expansions and extrapolations of eigenvalues for the stokes problem by mixed finite element methods,, J. Comput. Appl. Math., 215 (2008), 127.  doi: 10.1016/j.cam.2007.03.028.  Google Scholar [49] Z. Zhang, Y. Yang and Z. Chen, Eigenvalue approximation from below by Wilson's element,, (Chinese) Math. Numer. Sin., 29 (2007), 319.   Google Scholar
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