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 and 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), 109-117.

[2]

A. Andreev and T. Todorov, Isoparametric finite-element approximation of a Steklov eigenvalue problem, IMA J. Numer. Anal., 24 (2004), 309-322. doi: 10.1093/imanum/24.2.309.

[3]

M. Armentano and R. Durán, Asymptotic lower bounds for eigenvalues by nonconforming finite element methods, Electron. Trans. Numer. Anal., 17 (2004), 93-101.

[4]

I. Babuška and J. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comp., 52 (1989), 275-297. doi: 10.2307/2008468.

[5]

I. Babuška and J. Osborn, Eigenvalue Problems, in handbook of numerical analysis, Vol. II, 641-787, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991.

[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, Z. Angew. Math. Phys., 54 (2003), 874-878.

[7]

C. Bacuta, J. Bramble and J. Pasciak, "Shift Theorems for the Biharmonic Dirichlet Problem," Recent Progress in Computational and Appl. PDEs, proceedings of the International Symposium on Computational and Applied PDEs, Kluwer Academic/Plenum Publishers, 2001.

[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), 121-146. doi: 10.1006/jcph.1994.1182.

[9]

S. Bergman and M. Schiffer, "Kernel Functions and Elliptic Differential Equations in Mathematical Physics," Academic Press, New York, 1953.

[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), 201-227. doi: 10.1007/s002110000175.

[11]

J. Bramble and J. Osborn, Approximation of Steklov eigenvalues of non-selfadjoint second order elliptic operators, in "A. Aziz, (Ed.), Math. Foundations of the Finite Element Method with Applications to PDE," Academic, New York, (1972), 387-408.

[12]

F. Brezzi and M. Fortin, "Mixed and Hybrid Finite Element Methods," Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1.

[13]

D. Bucur and I. Ionescu, Asymptotic analysis and scaling of friction parameters, Z. Angew. Math. Phys. (ZAMP), 57 (2006), 1042-1056. doi: 10.1007/s00033-006-0070-9.

[14]

S. Brenner and L. Scott, "The Mathematical Theory of Finite Element Methods," Texts in Applied Mathematics, 15. Springer-Verlag, New York, 1994.

[15]

F. Chatelin, "Spectral Approximation of Linear Operators," With solutions to exercises by Mario Ahués. Computer Science and Applied Mathematics, Academic Press Inc, [Harcourt Brace Jovanovich, Publishers], New York, 1983.

[16]

P. Ciarlet, "The Finite Element Method for Elliptic Problem," Studies in Mathematics and its Applications, Vol. 4. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.

[17]

C. Conca, J. Planchard and M. Vanninathanm, "Fluid and Periodic Structures," John Wiley & Sons, New York, 1995.

[18]

K. Feng, A difference scheme based on variational principle, Appl. Math and Comp. Math, 2 (1965), 238-262.

[19]

V. Girault and P. Raviart, "Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms," Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[20]

P. Grisvard, "Singularities in Boundary Problems," MASSON and Springer-Verlag,, 1985., (). 

[21]

D. Hinton and J. Shaw, Differential operators with spectral parameter incompletely in the boundary conditions, Funkcialaj Ekvacioj (Serio Internacial), 33 (1990), 363-385.

[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), 545-556.

[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, \arxiv{1112.1145v1}. doi: 10.1007/s10915-013-9744-6.

[24]

E. Leriche and G. Labrosse, Stokes eigenmodes in square domain and the stream function-vorticity correlation, J. Comput. Phys., 200 (2004), 489-511. doi: 10.1016/j.jcp.2004.03.017.

[25]

Y. Li, Lower approximation of eigenvalue by the nonconforming finite element method, (Chinese) Math. Numer. Sin., 30 (2008), 195-200.

[26]

Y. Li, The lower bounds of eigenvalues by the Wilson element in any dimension, Adv. Appl. Math. Mech., 3 (2011), 598-610.

[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), 129-151. doi: 10.1007/s10492-013-0007-5.

[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), 2061-2084. doi: 10.1090/S0025-5718-08-02098-X.

[29]

Q. Lin and J. Lin, "Finite Element Methods: Accuracy and Improvement," Science Press: Beijing, 2006.

[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), 160-181. doi: 10.1093/imanum/drh008.

[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), 219-226.

[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), 157-168.

[33]

Q. Lin, H. Xie and J. Xu, Lower bounds of the discretization for piecewise polynomials, accepted by Math. Comp., http://arxiv.org/abs/1106.4395, June 22, (2011).

[34]

H. Liu and L. Liu, Expansion and extrapolation of the eigenvalue on $Q_1^{rot}$ element, Journal of Hebei University, 23 (2005), 11-15.

[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), 81-91.

[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), 1069-1082. doi: 10.1007/s11425-012-4382-2.

[37]

B. Mercier, J. Osborn, J. Rappaz and P. Raviart, Eigenvalue approximation by mixed and hybrid methods, Math. Comput., 36 (1981), 427-453. doi: 10.1090/S0025-5718-1981-0606505-9.

[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), 185-197. doi: 10.1137/0713019.

[39]

R. Rannacher, Nonconforming finite element methods for eigenvalue problems in linear plate theory, Numer. Math., 33 (1979), 23-42. doi: 10.1007/BF01396493.

[40]

G. Stang and G. Fix, "An Analysis of the Finite Element Method," Prentice-Hall Series in Automatic Computation. Prentice-Hall, Inc., Englewood Cliffs, NJ: Prentice-Hall, 1973.

[41]

Y. Yang, "Finite Element Methods Analysis to Eigenvalue Problem," Guizhou People Press: Guizhou, 2004.

[42]

Y. Yang and H. Bi, Lower spectral bounds by Wilson's brick discretization, Appl. Numer. Math., 60 (2010), 782-787. doi: 10.1016/j.apnum.2010.03.019.

[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), 1232-1242. doi: 10.1007/s11425-008-0002-6.

[44]

Y. Yang, Q. Li and S. Li, Nonconforming finite element approximations of the steklov eigenvalue problem, Appl. Numer. Math., 59 (2009), 2388-2401. doi: 10.1016/j.apnum.2009.04.005.

[45]

Y. Yang, Q. Lin, H. Bi and Q. Li, Eigenvalue approximations from below using Morley elements, Adv. Comput. Math., 36 (2012), 443-450. doi: 10.1007/s10444-011-9185-4.

[46]

Y. Yang, Z. Zhang and F. Lin, Eigenvalue approximation from below using nonforming finite elements, Sci. China Math., 53 (2010), 137-150. doi: 10.1007/s11425-009-0198-0.

[47]

C. Yao and Z. Qiao, Extrapolation of mixed finite element approximations for the maxwell eigenvalue problem, Numer. Math. Theory Methods Appl., 4 (2011), 379-395. doi: 10.4208/nmtma.2011.m1018.

[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), 127-141. doi: 10.1016/j.cam.2007.03.028.

[49]

Z. Zhang, Y. Yang and Z. Chen, Eigenvalue approximation from below by Wilson's element, (Chinese) Math. Numer. Sin., 29 (2007), 319-321.

show all references

References:
[1]

H. Ahn, Vibration of a pendulum consisiting of a bob suspended from a wire, Quart. Appl. Math., 39 (1981), 109-117.

[2]

A. Andreev and T. Todorov, Isoparametric finite-element approximation of a Steklov eigenvalue problem, IMA J. Numer. Anal., 24 (2004), 309-322. doi: 10.1093/imanum/24.2.309.

[3]

M. Armentano and R. Durán, Asymptotic lower bounds for eigenvalues by nonconforming finite element methods, Electron. Trans. Numer. Anal., 17 (2004), 93-101.

[4]

I. Babuška and J. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comp., 52 (1989), 275-297. doi: 10.2307/2008468.

[5]

I. Babuška and J. Osborn, Eigenvalue Problems, in handbook of numerical analysis, Vol. II, 641-787, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991.

[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, Z. Angew. Math. Phys., 54 (2003), 874-878.

[7]

C. Bacuta, J. Bramble and J. Pasciak, "Shift Theorems for the Biharmonic Dirichlet Problem," Recent Progress in Computational and Appl. PDEs, proceedings of the International Symposium on Computational and Applied PDEs, Kluwer Academic/Plenum Publishers, 2001.

[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), 121-146. doi: 10.1006/jcph.1994.1182.

[9]

S. Bergman and M. Schiffer, "Kernel Functions and Elliptic Differential Equations in Mathematical Physics," Academic Press, New York, 1953.

[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), 201-227. doi: 10.1007/s002110000175.

[11]

J. Bramble and J. Osborn, Approximation of Steklov eigenvalues of non-selfadjoint second order elliptic operators, in "A. Aziz, (Ed.), Math. Foundations of the Finite Element Method with Applications to PDE," Academic, New York, (1972), 387-408.

[12]

F. Brezzi and M. Fortin, "Mixed and Hybrid Finite Element Methods," Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1.

[13]

D. Bucur and I. Ionescu, Asymptotic analysis and scaling of friction parameters, Z. Angew. Math. Phys. (ZAMP), 57 (2006), 1042-1056. doi: 10.1007/s00033-006-0070-9.

[14]

S. Brenner and L. Scott, "The Mathematical Theory of Finite Element Methods," Texts in Applied Mathematics, 15. Springer-Verlag, New York, 1994.

[15]

F. Chatelin, "Spectral Approximation of Linear Operators," With solutions to exercises by Mario Ahués. Computer Science and Applied Mathematics, Academic Press Inc, [Harcourt Brace Jovanovich, Publishers], New York, 1983.

[16]

P. Ciarlet, "The Finite Element Method for Elliptic Problem," Studies in Mathematics and its Applications, Vol. 4. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.

[17]

C. Conca, J. Planchard and M. Vanninathanm, "Fluid and Periodic Structures," John Wiley & Sons, New York, 1995.

[18]

K. Feng, A difference scheme based on variational principle, Appl. Math and Comp. Math, 2 (1965), 238-262.

[19]

V. Girault and P. Raviart, "Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms," Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[20]

P. Grisvard, "Singularities in Boundary Problems," MASSON and Springer-Verlag,, 1985., (). 

[21]

D. Hinton and J. Shaw, Differential operators with spectral parameter incompletely in the boundary conditions, Funkcialaj Ekvacioj (Serio Internacial), 33 (1990), 363-385.

[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), 545-556.

[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, \arxiv{1112.1145v1}. doi: 10.1007/s10915-013-9744-6.

[24]

E. Leriche and G. Labrosse, Stokes eigenmodes in square domain and the stream function-vorticity correlation, J. Comput. Phys., 200 (2004), 489-511. doi: 10.1016/j.jcp.2004.03.017.

[25]

Y. Li, Lower approximation of eigenvalue by the nonconforming finite element method, (Chinese) Math. Numer. Sin., 30 (2008), 195-200.

[26]

Y. Li, The lower bounds of eigenvalues by the Wilson element in any dimension, Adv. Appl. Math. Mech., 3 (2011), 598-610.

[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), 129-151. doi: 10.1007/s10492-013-0007-5.

[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), 2061-2084. doi: 10.1090/S0025-5718-08-02098-X.

[29]

Q. Lin and J. Lin, "Finite Element Methods: Accuracy and Improvement," Science Press: Beijing, 2006.

[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), 160-181. doi: 10.1093/imanum/drh008.

[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), 219-226.

[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), 157-168.

[33]

Q. Lin, H. Xie and J. Xu, Lower bounds of the discretization for piecewise polynomials, accepted by Math. Comp., http://arxiv.org/abs/1106.4395, June 22, (2011).

[34]

H. Liu and L. Liu, Expansion and extrapolation of the eigenvalue on $Q_1^{rot}$ element, Journal of Hebei University, 23 (2005), 11-15.

[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), 81-91.

[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), 1069-1082. doi: 10.1007/s11425-012-4382-2.

[37]

B. Mercier, J. Osborn, J. Rappaz and P. Raviart, Eigenvalue approximation by mixed and hybrid methods, Math. Comput., 36 (1981), 427-453. doi: 10.1090/S0025-5718-1981-0606505-9.

[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), 185-197. doi: 10.1137/0713019.

[39]

R. Rannacher, Nonconforming finite element methods for eigenvalue problems in linear plate theory, Numer. Math., 33 (1979), 23-42. doi: 10.1007/BF01396493.

[40]

G. Stang and G. Fix, "An Analysis of the Finite Element Method," Prentice-Hall Series in Automatic Computation. Prentice-Hall, Inc., Englewood Cliffs, NJ: Prentice-Hall, 1973.

[41]

Y. Yang, "Finite Element Methods Analysis to Eigenvalue Problem," Guizhou People Press: Guizhou, 2004.

[42]

Y. Yang and H. Bi, Lower spectral bounds by Wilson's brick discretization, Appl. Numer. Math., 60 (2010), 782-787. doi: 10.1016/j.apnum.2010.03.019.

[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), 1232-1242. doi: 10.1007/s11425-008-0002-6.

[44]

Y. Yang, Q. Li and S. Li, Nonconforming finite element approximations of the steklov eigenvalue problem, Appl. Numer. Math., 59 (2009), 2388-2401. doi: 10.1016/j.apnum.2009.04.005.

[45]

Y. Yang, Q. Lin, H. Bi and Q. Li, Eigenvalue approximations from below using Morley elements, Adv. Comput. Math., 36 (2012), 443-450. doi: 10.1007/s10444-011-9185-4.

[46]

Y. Yang, Z. Zhang and F. Lin, Eigenvalue approximation from below using nonforming finite elements, Sci. China Math., 53 (2010), 137-150. doi: 10.1007/s11425-009-0198-0.

[47]

C. Yao and Z. Qiao, Extrapolation of mixed finite element approximations for the maxwell eigenvalue problem, Numer. Math. Theory Methods Appl., 4 (2011), 379-395. doi: 10.4208/nmtma.2011.m1018.

[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), 127-141. doi: 10.1016/j.cam.2007.03.028.

[49]

Z. Zhang, Y. Yang and Z. Chen, Eigenvalue approximation from below by Wilson's element, (Chinese) Math. Numer. Sin., 29 (2007), 319-321.

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