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May  2020, 25(5): 1775-1789. doi: 10.3934/dcdsb.2020002

The two-grid and multigrid discretizations of the $ C^0 $IPG method for biharmonic eigenvalue problem

Hao Li , Hai Bi and  Yidu Yang ,

School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550001, China

* Corresponding author: Yidu Yang

Received  May 2018 Revised  August 2019 Published  May 2020 Early access  December 2019

Fund Project: This work is supported by the National Natural Science Foundation of China (Grant No.11761022) and Science and Technology Foundation of Guizhou Province of China (Grant No. LH [2014] 7061).

In this paper, for the biharmonic eigenvalue problem with clamped boundary condition in $ \mathbb{R}^{2} $, we study the two-grid discretization based on shifted-inverse iteration of $ C^0 $IPG method. With our scheme, the solution of a biharmonic eigenvalue problem on a fine mesh $ \pi_h $ can be reduced to the solution of the eigenvalue problem on a coarser mesh $ \pi_H $ and the solution of a linear algebraic system on the fine mesh $ \pi_h $. We prove that the resulting solution still maintains an asymptotically optimal accuracy when $ h\geq O(H^3) $. In addition, we also discuss the multigrid discretization and the adaptive $ C^0 $IPG algorithm based on Rayleigh quotient iteration. Numerical experiments are provided to validate the theoretical analysis.

Keywords: The biharmonic eigenvalue problem, C0IPG method, two-grid discretization, Rayleigh quotient iteration, adaptive algorithm.
Mathematics Subject Classification: Primary: 65N25, 65N30.
Citation: Hao Li, Hai Bi, Yidu Yang. The two-grid and multigrid discretizations of the $ C^0 $IPG method for biharmonic eigenvalue problem. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1775-1789. doi: 10.3934/dcdsb.2020002
References:
[1]

M. Ainsworth and J. T. Oden, A Posteriori Error Estimates in the Finite Element Analysis, Wiley-Inter science, New York, 2000. doi: 10.1002/9781118032824.

[2]

A. B. Andreev, R. D. Lazarov and M. R. Racheva, Postprocessing and higher order convergence of the mixed finite element approximations of biharmonic eigenvalue problems, J. Comput. Appl. Math., 182 (2005), 333-349.  doi: 10.1016/j.cam.2004.12.015.

[3]

I. Babuška, R. B. Kellog and J. Pitkaranta, Direct and inverse error estimates for finite elements with mesh refinement, Numer. Math., 33 (1979), 447-471.  doi: 10.1007/BF01399326.

[4]

I. Babuška and J. Osborn, Eigenvalue problems, Handbook of Numerical Analysis, Handb. Numer. Anal., North-Holland, Amsterdam, 2 (1991), 641-787. 

[5]

I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15 (1978), 736-754.  doi: 10.1137/0715049.

[6]

L. Beirão da Veiga, J. Niiranen and R. Stenberg, A posteriori erros estimates for the Morley plate bending element, Numer. Math., 106 (2007), 165-179.  doi: 10.1007/s00211-007-0066-1.

[7]

H. Blum and R. Rannacher, On the boundary value problem of the biharmonic operator on domains with angular corners, Math. Method Appl. Sci., 2 (1980), 556-581.  doi: 10.1002/mma.1670020416.

[8]

D. Boffi, Finite element approximation of eigenvalue problems, Acta Numerica, 19 (2010), 1-120.  doi: 10.1017/S0962492910000012.

[9]

S. C. Brenner, $C^{0}$ interior penalty methods, Frontiers in Numerical Analysis-Durham 2010, Lecture Notes in Computational Science and Engineering, Springer-Verlag, 85 (2012), 79-147.  doi: 10.1007/978-3-642-23914-4_2.

[10]

S. C. Brenner and et al., Adaptive $C^0$ interior penalty method for biharmonic eigenvalue problems, Numerical Solution of PDE Eigenvalue Problems, Oberwolfach Rep, 10 (2013), 3265-3267. 

[11]

S. C. Brenner, S. Y. Gu, T. Gudi and L.-Y. Sung, A quadratic $C^{0}$ interior penalty method for linear fourth order boundary value problems with boundary conditions of the Cahn-Hilliard type, SIAM J. Numer. Anal., 50 (2012), 2088-2110.  doi: 10.1137/110847469.

[12]

S. C. Brenner, P. Monk and J. G. Sun, $C^{0}$IPG method for biharmonic eigenvalue problems, Spectral and High Order Methods for Partial Differential Equation-ICOSAHOM 2014, Lect. Notes Comput. Sci. Eng., Springer, Cham, 106 (2015), 3-15. 

[13]

S. C. Brenner and M. Neilan, A $C^{0}$ interior penalty method for a fourth order elliptic singular perturbation problem, SIAM J. Numer. Anal., 49 (2011), 869-892.  doi: 10.1137/100786988.

[14]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Second edition, Texts in Applied Mathematics, 15. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-3658-8.

[15]

S. C. Brenner and L.-Y. Sung, $C^{0}$ interior penalty methods for fourth order elliptic boundary value problems on polygonal domains, J. Sci. Comput., 22/23 (2005), 83-118.  doi: 10.1007/s10915-004-4135-7.

[16]

S. C. Brenner, K. N. Wang and J. Zhao, Poincaré-Friedrichs inequalities for piecewise $H^{2}$ functions, Numer. Funct. Anal. Optim., 25 (2004), 463-478.  doi: 10.1081/NFA-200042165.

[17]

C. Carstensen and D. Gallistl, Guaranteed lower eigenvalue bounds for the biharmonic equation, Numer. Math., 125 (2014), 33-51.  doi: 10.1007/s00211-013-0559-z.

[18] F. Chatelin, Spectral Approximations of Linear Operators, Computer Science and Applied Mathematics, Academic Press, Inc., New York, 1983. 
[19]

H. T. Chen, H. L. Guo, Z. M. Zhang and Q. S. Zou, A $C^0$ linear finite element method for two fourth-order eignvalue problems, IMA J. Numer. Anal., 37 (2017), 2120-2138.  doi: 10.1093/imanum/drw051.

[20]

L. Chen, IFEM: An Innovative Finite Element Methods Package in MATLAB, Technical Report, University of California at Irvine, 2009.

[21]

P. G. Ciarlet, Basic error estimates for elliptic proplems, Handbook of Numerical Analysis, Handb. Numer. Anal., Ⅱ, North-Holland, Amsterdam, 2 (1991), 17-351. 

[22]

X. Y. Dai, J. C. Xu and A. H. Zhou, Convergence and optimal complexity of adaptive finite element eigenvalue computations, Numer. Math., 110 (2008), 313-355.  doi: 10.1007/s00211-008-0169-3.

[23]

X. Y. Dai and A. H. Zhou, Three-scale finite element discretizations for quantum eigenvalue problems, SIAM J. Numer. Anal., 46 (2007/08), 295-324.  doi: 10.1137/06067780X.

[24]

G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. Mazzei and R. L. Taylor, Continuous/discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Comput. Methods Appl. Mech. Engrg., 191 (2002), 3669-3750.  doi: 10.1016/S0045-7825(02)00286-4.

[25]

H. R. Geng, X. Ji, J. G. Sun and L. W. Xu, $C^{0}$IP methods for the transmission eigenvalue problem, J. Sci. Comput., 68 (2016), 326-338.  doi: 10.1007/s10915-015-0140-2.

[26]

T. Gudi, A new error analysis for discontinuous finite element methods for the linear elliptic problems, Math. Comp., 79 (2010), 2169-2189.  doi: 10.1090/S0025-5718-10-02360-4.

[27]

H. L. Guo, Z. M. Zhang and Q. S. Zou, A $C^0$ linear finite element method for biharmonic problems, J. Sci. Comput., 74 (2018), 1397-1422.  doi: 10.1007/s10915-017-0501-0.

[28]

H. L. Guo, Z. M. Zhang and R. Zhao, Superconvergent two-grid schemes for elliptic eigenvalue problems, J. Sci. Comput., 70 (2017), 125-148.  doi: 10.1007/s10915-016-0245-2.

[29]

X. Z. Hu and X. L. Cheng, Acceleration of a two-grid method for eigenvalue problems, Math. Comp., 80 (2011), 1287-1301.  doi: 10.1090/S0025-5718-2011-02458-0.

[30]

J. Hu, Y. Q. Huang and Q. Lin, Lower bounds for eigenvalues of elliptic operators: By Nonconforming finite element methods, J. Sci. Comput., 61 (2014), 196-221.  doi: 10.1007/s10915-014-9821-5.

[31]

J. Hu, Z. C. Shi and J. C. Xu, Convergence and optimality of the adaptive Morley element method, Numer. Math., 121 (2012), 731-752.  doi: 10.1007/s00211-012-0445-0.

[32]

H. Li and Y. D. Yang, $C^0$IPG adaptive algorithms for the biharmonic eigenvalue problem, Numer. Algor., 78 (2018), 553-567.  doi: 10.1007/s11075-017-0388-8.

[33]

H. Li and Y. D. Yang, An adaptive $C^0$IPG method for the Helmholtz transmission eigenvalue problem, Science China Mathematics, 61 (2018), 1519-1542.  doi: 10.1007/s11425-017-9334-9.

[34] Q. Lin and J. Lin, Finite Element Methods: Accuracy and Inprovement, Science Press, Beijing, 2006. 
[35]

Q. Lin and H. H. Xie, A multi-level correction scheme for eigenvalue problems, Math. Comp., 84 (2015), 71-88.  doi: 10.1090/S0025-5718-2014-02825-1.

[36]

P. Morin, R. H. Nochetto and K. G. Siebert, Convergence of adaptive finite element methods, SIAM Rev., 44 (2002), 631-658.  doi: 10.1137/S0036144502409093.

[37]

J. T. Oden and J. N. Reddy, An Introduction to the Mathematical Theory of Finite Elements, Pure and Applied Mathematics. Wiley-Interscience, New York-London-Sydney, 1976.

[38]

Q. Shen, A posteriori error estimates of the Morley element for the fourth order elliptic eigenvalue problem, Numer. Algor., 68 (2015), 455-466.  doi: 10.1007/s11075-014-9854-8.

[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] Z. Shi and M. Wang, Finite Element Methods, Science Press, Beijing, 2013. 
[41]

R. Verfürth, A Review of a Posteriori Error Estimates and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, New York, 1996.

[42]

G. N. Wells and N. T. Dung, A $C^{0}$ discontinuous Galerkin formulation for Kirhhoff plates, Comput. Methods Appl. Mech. Engrg., 196 (2007), 3370-3380.  doi: 10.1016/j.cma.2007.03.008.

[43]

H. H. Xie and X. B. Yin, Acceleration of stabilized finite element discretizations for the Stokes eigenvalue problem, Adv. Comput. Math., 41 (2015), 799-812.  doi: 10.1007/s10444-014-9386-8.

[44]

J. C. Xu, A new class of iterative methods for nonselfadjoint or indefinite problems, SIAM J. Numer. Anal., 29 (1992), 303-319.  doi: 10.1137/0729020.

[45]

J. C. Xu and A. H. Zhou, A two-grid discretization scheme for eigenvalue problems, Math. Comput., 70 (2001), 17-25.  doi: 10.1090/S0025-5718-99-01180-1.

[46]

Y. D. Yang and H. Bi, Two-grid finite element discretization scheme based on shifted-inverse power method for elliptic eigenvalue problems, SIAM J. Numer. Anal., 49 (2011), 1602-1624.  doi: 10.1137/100810241.

[47]

Y. D. Yang, H. Bi, J. Y. Han and Y. Y. Yu, The shifted-inverse iteration based on the multigrid discertiaztions for eigenvalue problems, SIAM J. Sci. Comput., 37 (2015), A2583–A2606. doi: 10.1137/140992011.

[48]

Y. D. Yang, H. Bi, H. Li and J. Y. Han, A $C^{0}$IPG method and its error estimates for the Helmholtz transmission eigenvalue problem, J. Comput. Appl. Math., 326 (2017), 71-86.  doi: 10.1016/j.cam.2017.04.024.

[49]

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

[50]

J. Zhou, X. Hu, L. Zhong, S. Shu and L. Chen, Two-grid methods for Maxwell eigenvalue problems, SIAM J. Numer. Anal., 52 (2014), 2027-2047.  doi: 10.1137/130919921.

[51]

O. C. Zienkiewicz, The Finite Element Method in Engineering Science, McGraw-Hill, London-New York-Düsseldorf, 1971.

show all references

References:
[1]

M. Ainsworth and J. T. Oden, A Posteriori Error Estimates in the Finite Element Analysis, Wiley-Inter science, New York, 2000. doi: 10.1002/9781118032824.

[2]

A. B. Andreev, R. D. Lazarov and M. R. Racheva, Postprocessing and higher order convergence of the mixed finite element approximations of biharmonic eigenvalue problems, J. Comput. Appl. Math., 182 (2005), 333-349.  doi: 10.1016/j.cam.2004.12.015.

[3]

I. Babuška, R. B. Kellog and J. Pitkaranta, Direct and inverse error estimates for finite elements with mesh refinement, Numer. Math., 33 (1979), 447-471.  doi: 10.1007/BF01399326.

[4]

I. Babuška and J. Osborn, Eigenvalue problems, Handbook of Numerical Analysis, Handb. Numer. Anal., North-Holland, Amsterdam, 2 (1991), 641-787. 

[5]

I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15 (1978), 736-754.  doi: 10.1137/0715049.

[6]

L. Beirão da Veiga, J. Niiranen and R. Stenberg, A posteriori erros estimates for the Morley plate bending element, Numer. Math., 106 (2007), 165-179.  doi: 10.1007/s00211-007-0066-1.

[7]

H. Blum and R. Rannacher, On the boundary value problem of the biharmonic operator on domains with angular corners, Math. Method Appl. Sci., 2 (1980), 556-581.  doi: 10.1002/mma.1670020416.

[8]

D. Boffi, Finite element approximation of eigenvalue problems, Acta Numerica, 19 (2010), 1-120.  doi: 10.1017/S0962492910000012.

[9]

S. C. Brenner, $C^{0}$ interior penalty methods, Frontiers in Numerical Analysis-Durham 2010, Lecture Notes in Computational Science and Engineering, Springer-Verlag, 85 (2012), 79-147.  doi: 10.1007/978-3-642-23914-4_2.

[10]

S. C. Brenner and et al., Adaptive $C^0$ interior penalty method for biharmonic eigenvalue problems, Numerical Solution of PDE Eigenvalue Problems, Oberwolfach Rep, 10 (2013), 3265-3267. 

[11]

S. C. Brenner, S. Y. Gu, T. Gudi and L.-Y. Sung, A quadratic $C^{0}$ interior penalty method for linear fourth order boundary value problems with boundary conditions of the Cahn-Hilliard type, SIAM J. Numer. Anal., 50 (2012), 2088-2110.  doi: 10.1137/110847469.

[12]

S. C. Brenner, P. Monk and J. G. Sun, $C^{0}$IPG method for biharmonic eigenvalue problems, Spectral and High Order Methods for Partial Differential Equation-ICOSAHOM 2014, Lect. Notes Comput. Sci. Eng., Springer, Cham, 106 (2015), 3-15. 

[13]

S. C. Brenner and M. Neilan, A $C^{0}$ interior penalty method for a fourth order elliptic singular perturbation problem, SIAM J. Numer. Anal., 49 (2011), 869-892.  doi: 10.1137/100786988.

[14]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Second edition, Texts in Applied Mathematics, 15. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-3658-8.

[15]

S. C. Brenner and L.-Y. Sung, $C^{0}$ interior penalty methods for fourth order elliptic boundary value problems on polygonal domains, J. Sci. Comput., 22/23 (2005), 83-118.  doi: 10.1007/s10915-004-4135-7.

[16]

S. C. Brenner, K. N. Wang and J. Zhao, Poincaré-Friedrichs inequalities for piecewise $H^{2}$ functions, Numer. Funct. Anal. Optim., 25 (2004), 463-478.  doi: 10.1081/NFA-200042165.

[17]

C. Carstensen and D. Gallistl, Guaranteed lower eigenvalue bounds for the biharmonic equation, Numer. Math., 125 (2014), 33-51.  doi: 10.1007/s00211-013-0559-z.

[18] F. Chatelin, Spectral Approximations of Linear Operators, Computer Science and Applied Mathematics, Academic Press, Inc., New York, 1983. 
[19]

H. T. Chen, H. L. Guo, Z. M. Zhang and Q. S. Zou, A $C^0$ linear finite element method for two fourth-order eignvalue problems, IMA J. Numer. Anal., 37 (2017), 2120-2138.  doi: 10.1093/imanum/drw051.

[20]

L. Chen, IFEM: An Innovative Finite Element Methods Package in MATLAB, Technical Report, University of California at Irvine, 2009.

[21]

P. G. Ciarlet, Basic error estimates for elliptic proplems, Handbook of Numerical Analysis, Handb. Numer. Anal., Ⅱ, North-Holland, Amsterdam, 2 (1991), 17-351. 

[22]

X. Y. Dai, J. C. Xu and A. H. Zhou, Convergence and optimal complexity of adaptive finite element eigenvalue computations, Numer. Math., 110 (2008), 313-355.  doi: 10.1007/s00211-008-0169-3.

[23]

X. Y. Dai and A. H. Zhou, Three-scale finite element discretizations for quantum eigenvalue problems, SIAM J. Numer. Anal., 46 (2007/08), 295-324.  doi: 10.1137/06067780X.

[24]

G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. Mazzei and R. L. Taylor, Continuous/discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Comput. Methods Appl. Mech. Engrg., 191 (2002), 3669-3750.  doi: 10.1016/S0045-7825(02)00286-4.

[25]

H. R. Geng, X. Ji, J. G. Sun and L. W. Xu, $C^{0}$IP methods for the transmission eigenvalue problem, J. Sci. Comput., 68 (2016), 326-338.  doi: 10.1007/s10915-015-0140-2.

[26]

T. Gudi, A new error analysis for discontinuous finite element methods for the linear elliptic problems, Math. Comp., 79 (2010), 2169-2189.  doi: 10.1090/S0025-5718-10-02360-4.

[27]

H. L. Guo, Z. M. Zhang and Q. S. Zou, A $C^0$ linear finite element method for biharmonic problems, J. Sci. Comput., 74 (2018), 1397-1422.  doi: 10.1007/s10915-017-0501-0.

[28]

H. L. Guo, Z. M. Zhang and R. Zhao, Superconvergent two-grid schemes for elliptic eigenvalue problems, J. Sci. Comput., 70 (2017), 125-148.  doi: 10.1007/s10915-016-0245-2.

[29]

X. Z. Hu and X. L. Cheng, Acceleration of a two-grid method for eigenvalue problems, Math. Comp., 80 (2011), 1287-1301.  doi: 10.1090/S0025-5718-2011-02458-0.

[30]

J. Hu, Y. Q. Huang and Q. Lin, Lower bounds for eigenvalues of elliptic operators: By Nonconforming finite element methods, J. Sci. Comput., 61 (2014), 196-221.  doi: 10.1007/s10915-014-9821-5.

[31]

J. Hu, Z. C. Shi and J. C. Xu, Convergence and optimality of the adaptive Morley element method, Numer. Math., 121 (2012), 731-752.  doi: 10.1007/s00211-012-0445-0.

[32]

H. Li and Y. D. Yang, $C^0$IPG adaptive algorithms for the biharmonic eigenvalue problem, Numer. Algor., 78 (2018), 553-567.  doi: 10.1007/s11075-017-0388-8.

[33]

H. Li and Y. D. Yang, An adaptive $C^0$IPG method for the Helmholtz transmission eigenvalue problem, Science China Mathematics, 61 (2018), 1519-1542.  doi: 10.1007/s11425-017-9334-9.

[34] Q. Lin and J. Lin, Finite Element Methods: Accuracy and Inprovement, Science Press, Beijing, 2006. 
[35]

Q. Lin and H. H. Xie, A multi-level correction scheme for eigenvalue problems, Math. Comp., 84 (2015), 71-88.  doi: 10.1090/S0025-5718-2014-02825-1.

[36]

P. Morin, R. H. Nochetto and K. G. Siebert, Convergence of adaptive finite element methods, SIAM Rev., 44 (2002), 631-658.  doi: 10.1137/S0036144502409093.

[37]

J. T. Oden and J. N. Reddy, An Introduction to the Mathematical Theory of Finite Elements, Pure and Applied Mathematics. Wiley-Interscience, New York-London-Sydney, 1976.

[38]

Q. Shen, A posteriori error estimates of the Morley element for the fourth order elliptic eigenvalue problem, Numer. Algor., 68 (2015), 455-466.  doi: 10.1007/s11075-014-9854-8.

[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] Z. Shi and M. Wang, Finite Element Methods, Science Press, Beijing, 2013. 
[41]

R. Verfürth, A Review of a Posteriori Error Estimates and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, New York, 1996.

[42]

G. N. Wells and N. T. Dung, A $C^{0}$ discontinuous Galerkin formulation for Kirhhoff plates, Comput. Methods Appl. Mech. Engrg., 196 (2007), 3370-3380.  doi: 10.1016/j.cma.2007.03.008.

[43]

H. H. Xie and X. B. Yin, Acceleration of stabilized finite element discretizations for the Stokes eigenvalue problem, Adv. Comput. Math., 41 (2015), 799-812.  doi: 10.1007/s10444-014-9386-8.

[44]

J. C. Xu, A new class of iterative methods for nonselfadjoint or indefinite problems, SIAM J. Numer. Anal., 29 (1992), 303-319.  doi: 10.1137/0729020.

[45]

J. C. Xu and A. H. Zhou, A two-grid discretization scheme for eigenvalue problems, Math. Comput., 70 (2001), 17-25.  doi: 10.1090/S0025-5718-99-01180-1.

[46]

Y. D. Yang and H. Bi, Two-grid finite element discretization scheme based on shifted-inverse power method for elliptic eigenvalue problems, SIAM J. Numer. Anal., 49 (2011), 1602-1624.  doi: 10.1137/100810241.

[47]

Y. D. Yang, H. Bi, J. Y. Han and Y. Y. Yu, The shifted-inverse iteration based on the multigrid discertiaztions for eigenvalue problems, SIAM J. Sci. Comput., 37 (2015), A2583–A2606. doi: 10.1137/140992011.

[48]

Y. D. Yang, H. Bi, H. Li and J. Y. Han, A $C^{0}$IPG method and its error estimates for the Helmholtz transmission eigenvalue problem, J. Comput. Appl. Math., 326 (2017), 71-86.  doi: 10.1016/j.cam.2017.04.024.

[49]

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

[50]

J. Zhou, X. Hu, L. Zhong, S. Shu and L. Chen, Two-grid methods for Maxwell eigenvalue problems, SIAM J. Numer. Anal., 52 (2014), 2027-2047.  doi: 10.1137/130919921.

[51]

O. C. Zienkiewicz, The Finite Element Method in Engineering Science, McGraw-Hill, London-New York-Düsseldorf, 1971.

Figure 1.  The convergence rates for the unit square with a slit using quadratic(left) and cubic(right) $ C^0 $IPG methods
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Figure 2.  The convergence rates for the L-shaped domain using quadratic(left) and cubic(right) $ C^0 $IPG methods
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Table 1.  the first eigenvalue approximation for (2.1) on the unit square using quadratic $ C^0 $IPG method
$ H $ $ h $ $ \lambda_{H} $ $ t_1(s) $ $ \lambda^h $ $ t_2(s) $ $ \lambda_{h} $ $ t_3(s) $
$ \frac{\sqrt{2}}{8} $ $ \frac{\sqrt{2}}{32} $ 1570.1117 .013 1307.7056 .033 1307.7017 .058
$ \frac{\sqrt{2}}{16} $ $ \frac{\sqrt{2}}{128} $ 1350.9328 .044 1295.6742 1.01 1295.6742 1.61
$ \frac{\sqrt{2}}{32} $ $ \frac{\sqrt{2}}{512} $ 1307.7017 .055 1294.9765 37.9 1294.9736 56.9
$ \frac{\sqrt{2}}{64} $ $ \frac{\sqrt{2}}{1024} $ 1297.9745 .348 1294.8984 283 $ – $ $ – $
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$ H $ $ h $ $ \lambda_{H} $ $ t_1(s) $ $ \lambda^h $ $ t_2(s) $ $ \lambda_{h} $ $ t_3(s) $
$ \frac{\sqrt{2}}{8} $ $ \frac{\sqrt{2}}{32} $ 1570.1117 .013 1307.7056 .033 1307.7017 .058
$ \frac{\sqrt{2}}{16} $ $ \frac{\sqrt{2}}{128} $ 1350.9328 .044 1295.6742 1.01 1295.6742 1.61
$ \frac{\sqrt{2}}{32} $ $ \frac{\sqrt{2}}{512} $ 1307.7017 .055 1294.9765 37.9 1294.9736 56.9
$ \frac{\sqrt{2}}{64} $ $ \frac{\sqrt{2}}{1024} $ 1297.9745 .348 1294.8984 283 $ – $ $ – $
Table 2.  the first eigenvalue approximation for (2.1) on the unit square using cubic $ C^0 $IPG method
$ H $ $ h $ $ \lambda_{H} $ $ t_1(s) $ $ \lambda^h $ $ t_2(s) $ $ \lambda_{h} $ $ t_3(s) $
$ \frac{\sqrt{2}}{8} $ $ \frac{\sqrt{2}}{32} $ 1300.7399 .017 1294.9569 .111 1294.9569 .226
$ \frac{\sqrt{2}}{16} $ $ \frac{\sqrt{2}}{128} $ 1295.3239 .069 1294.9322 3.83 1294.9322 6.93
$ \frac{\sqrt{2}}{32} $ $ \frac{\sqrt{2}}{512} $ 1294.9569 .222 1294.4529 208 1294.4621 1602
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$ H $ $ h $ $ \lambda_{H} $ $ t_1(s) $ $ \lambda^h $ $ t_2(s) $ $ \lambda_{h} $ $ t_3(s) $
$ \frac{\sqrt{2}}{8} $ $ \frac{\sqrt{2}}{32} $ 1300.7399 .017 1294.9569 .111 1294.9569 .226
$ \frac{\sqrt{2}}{16} $ $ \frac{\sqrt{2}}{128} $ 1295.3239 .069 1294.9322 3.83 1294.9322 6.93
$ \frac{\sqrt{2}}{32} $ $ \frac{\sqrt{2}}{512} $ 1294.9569 .222 1294.4529 208 1294.4621 1602
Table 3.  the first eigenvalue approximation for (2.1) on the unit square with a slit using quadratic $ C^0 $IPG method
$ H $ $ h $ $ \lambda_{H} $ $ t_1(s) $ $ \lambda^h $ $ t_2(s) $ $ \lambda_{h} $ $ t_3(s) $
$ \frac{\sqrt{2}}{8} $ $ \frac{\sqrt{2}}{32} $ 10678.0553 .010 6541.4871 .037 6539.8146 .051
$ \frac{\sqrt{2}}{16} $ $ \frac{\sqrt{2}}{128} $ 7228.7805 .011 6250.7646 .948 6250.6718 1.42
$ \frac{\sqrt{2}}{32} $ $ \frac{\sqrt{2}}{512} $ 6539.8146 .070 6202.8719 29.7 6202.8660 44.0
$ \frac{\sqrt{2}}{64} $ $ \frac{\sqrt{2}}{1024} $ 6328.9667 .298 6195.7389 217 $ -- $ $ -- $
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$ H $ $ h $ $ \lambda_{H} $ $ t_1(s) $ $ \lambda^h $ $ t_2(s) $ $ \lambda_{h} $ $ t_3(s) $
$ \frac{\sqrt{2}}{8} $ $ \frac{\sqrt{2}}{32} $ 10678.0553 .010 6541.4871 .037 6539.8146 .051
$ \frac{\sqrt{2}}{16} $ $ \frac{\sqrt{2}}{128} $ 7228.7805 .011 6250.7646 .948 6250.6718 1.42
$ \frac{\sqrt{2}}{32} $ $ \frac{\sqrt{2}}{512} $ 6539.8146 .070 6202.8719 29.7 6202.8660 44.0
$ \frac{\sqrt{2}}{64} $ $ \frac{\sqrt{2}}{1024} $ 6328.9667 .298 6195.7389 217 $ -- $ $ -- $
Table 4.  the first eigenvalue approximation for (2.1) on the unit square with a slit using cubic $ C^0 $IPG method
$ H $ $ h $ $ \lambda_{H} $ $ t_1(s) $ $ \lambda^h $ $ t_2(s) $ $ \lambda_{h} $ $ t_3(s) $
$ \frac{\sqrt{2}}{8} $ $ \frac{\sqrt{2}}{32} $ 6416.8901 .022 6226.2450 .096 6226.2445 .192
$ \frac{\sqrt{2}}{16} $ $ \frac{\sqrt{2}}{128} $ 6271.1319 .032 6197.8931 3.24 6197.8930 5.91
$ \frac{\sqrt{2}}{32} $ $ \frac{\sqrt{2}}{512} $ 6226.2445 .192 6190.6868 165 6190.6738 216
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$ H $ $ h $ $ \lambda_{H} $ $ t_1(s) $ $ \lambda^h $ $ t_2(s) $ $ \lambda_{h} $ $ t_3(s) $
$ \frac{\sqrt{2}}{8} $ $ \frac{\sqrt{2}}{32} $ 6416.8901 .022 6226.2450 .096 6226.2445 .192
$ \frac{\sqrt{2}}{16} $ $ \frac{\sqrt{2}}{128} $ 6271.1319 .032 6197.8931 3.24 6197.8930 5.91
$ \frac{\sqrt{2}}{32} $ $ \frac{\sqrt{2}}{512} $ 6226.2445 .192 6190.6868 165 6190.6738 216
Table 5.  the first eigenvalue approximation for (2.1) on the L-shaped domain using quadratic $ C^0 $IPG method
$ H $ $ h $ $ \lambda_{H} $ $ t_1(s) $ $ \lambda^h $ $ t_2(s) $ $ \lambda_{h} $ $ t_3(s) $
$ \frac{\sqrt{2}}{8} $ $ \frac{\sqrt{2}}{32} $ 11346.3219 .008 7019.4044 .017 7017.9070 .031
$ \frac{\sqrt{2}}{16} $ $ \frac{\sqrt{2}}{128} $ 7713.4188 .008 6750.3026 .613 6750.2705 1.13
$ \frac{\sqrt{2}}{32} $ $ \frac{\sqrt{2}}{512} $ 7017.9070 .032 6712.6852 21.8 6712.6814 32.3
$ \frac{\sqrt{2}}{64} $ $ \frac{\sqrt{2}}{1024} $ 6818.6359 .204 6707.7302 155 6707.6827 191
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$ H $ $ h $ $ \lambda_{H} $ $ t_1(s) $ $ \lambda^h $ $ t_2(s) $ $ \lambda_{h} $ $ t_3(s) $
$ \frac{\sqrt{2}}{8} $ $ \frac{\sqrt{2}}{32} $ 11346.3219 .008 7019.4044 .017 7017.9070 .031
$ \frac{\sqrt{2}}{16} $ $ \frac{\sqrt{2}}{128} $ 7713.4188 .008 6750.3026 .613 6750.2705 1.13
$ \frac{\sqrt{2}}{32} $ $ \frac{\sqrt{2}}{512} $ 7017.9070 .032 6712.6852 21.8 6712.6814 32.3
$ \frac{\sqrt{2}}{64} $ $ \frac{\sqrt{2}}{1024} $ 6818.6359 .204 6707.7302 155 6707.6827 191
Table 6.  the first eigenvalue approximation for (2.1) on the L-shaped domain using cubic $ C^0 $IPG method
$ H $ $ h $ $ \lambda_{H} $ $ t_1(s) $ $ \lambda^h $ $ t_2(s) $ $ \lambda_{h} $ $ t_3(s) $
$ \frac{\sqrt{2}}{8} $ $ \frac{\sqrt{2}}{32} $ 6896.1972 .009 6729.3265 .066 6729.3263 .136
$ \frac{\sqrt{2}}{16} $ $ \frac{\sqrt{2}}{128} $ 6764.7073 .022 6709.0991 2.35 6709.0990 4.16
$ \frac{\sqrt{2}}{32} $ $ \frac{\sqrt{2}}{512} $ 6729.3263 .136 6704.3362 117 6704.3206 157
Table Options

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$ H $ $ h $ $ \lambda_{H} $ $ t_1(s) $ $ \lambda^h $ $ t_2(s) $ $ \lambda_{h} $ $ t_3(s) $
$ \frac{\sqrt{2}}{8} $ $ \frac{\sqrt{2}}{32} $ 6896.1972 .009 6729.3265 .066 6729.3263 .136
$ \frac{\sqrt{2}}{16} $ $ \frac{\sqrt{2}}{128} $ 6764.7073 .022 6709.0991 2.35 6709.0990 4.16
$ \frac{\sqrt{2}}{32} $ $ \frac{\sqrt{2}}{512} $ 6729.3263 .136 6704.3362 117 6704.3206 157
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