doi: 10.3934/dcdsb.2022066
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A novel numerical method based on a high order polynomial approximation of the fourth order Steklov equation and its eigenvalue problems

1. 

School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550025, China

2. 

School of Mathematics and Statistics, Linyi University, Linyi, 276005, China

*Corresponding author: Jing An and Jianwei Zhou

Received  December 2021 Revised  February 2022 Early access March 2022

Fund Project: This work is supported by the National Natural Science Foundation of China (Grant No.12061023), and Guizhou Provincial Education Department Foundation (Qianjiaohe No.KY[2018]041)

Based on high order polynomial approximation and dimension reduction technique, we propose a novel numerical method for the fourth order Steklov problems in the circular domain. We first decompose the primal problem into a set of 1D problems via polar coordinate transformation and Fourier basis functions expansion. Then, by introducing a non-uniformly weighed Sobolev space, the variational form and corresponding discrete scheme are derived. Employing the Lax-Milgram lemma and approximation properties of the projection operators, we further prove existence and uniqueness of weak solutions and approximation solutions for each one-dimensional problems, and the error estimation between them, respectively. We also carry out ample numerical experiments which illustrate that the numerical algorithm is efficient and highly accurate.

Citation: Jiantao Jiang, Jing An, Jianwei Zhou. A novel numerical method based on a high order polynomial approximation of the fourth order Steklov equation and its eigenvalue problems. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022066
References:
[1]

J. AnH. Bi and Z. Luo, A highly efficient spectral-Galerkin method based on tensor product for fourth-order Steklov equation with boundary eigenvalue, J. Inequal. Appl., 2016 (2016), 1-12.  doi: 10.1186/s13660-016-1158-1.

[2]

J. AnH. Li and Z. Zhang, Spectral-Galerkin approximation and optimal error estimate for biharmonic eigenvalue problems in circular/spherical/elliptical domains, Numer. Algorithms, 84 (2020), 427-455.  doi: 10.1007/s11075-019-00760-4.

[3]

J. An and Z. Zhang, An efficient spectral-Galerkin approximation and error analysis for Maxwell transmission eigenvalue problems in spherical geometries, J. Sci. Comput., 75 (2018), 157-181.  doi: 10.1007/s10915-017-0528-2.

[4]

A. B. Andreev and T. D. 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.

[5] S. Bergman and M. Schiffer, Kernel Functions and Elliptic Differential Equations in Mathematical Physics, Academic Press, Inc., New York, N.Y., 1953. 
[6]

E. BerchioF. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems, J. Differential Equations, 229 (2006), 1-23.  doi: 10.1016/j.jde.2006.04.003.

[7]

A. BermúdezR. Rodríguez 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.

[8]

H. BiH. Li and Y. Yang, An adaptive algorithm based on the shifted inverse iteration for the Steklov eigenvalue problem, Appl. Numer. Math., 105 (2016), 64-81.  doi: 10.1016/j.apnum.2016.02.003.

[9]

H. Bi, S. Ren and Y. Yang, Conforming finite element approximations for a fourth-order Steklov eigenvalue problem, Math. Probl. Eng., 2011 (2011), Art. ID 873152, 13 pp. doi: 10.1155/2011/873152.

[10]

H. Bi and Y. Yang, A two-grid method of the non-conforming crouzeix–raviart element for the Steklov eigenvalue problem, Appl. Math. Comput., 217 (2011), 9669-9678.  doi: 10.1016/j.amc.2011.04.051.

[11]

D. BucurA. Ferrero and F. Gazzola, On the first eigenvalue of a fourth order Steklov problem, Calc. Var. Partial Differential Equations, 35 (2009), 103-131.  doi: 10.1007/s00526-008-0199-9.

[12]

D. Bucur and F. Gazzola, The first biharmonic Steklov eigenvalue: Positivity preserving and shape optimization, Milan J. Math., 79 (2011), 247-258.  doi: 10.1007/s00032-011-0143-x.

[13]

C. Conca, J. Planchard and M. Vanninathan, Fluids and Periodic Structures, John Wiley and Sons, Ltd., Chichester; Masson, Paris, 1995.

[14]

A. FerreroF. Gazzola and T. Weth, On a fourth order Steklov eigenvalue problem, Analysis, 25 (2005), 315-332.  doi: 10.1524/anly.2005.25.4.315.

[15]

F. Gazzola and G. Sweers, On positivity for the biharmonic operator under Steklov boundary conditions, Arch. Ration. Mech. Anal., 188 (2008), 399-427.  doi: 10.1007/s00205-007-0090-4.

[16]

X. HanY. Li and H. Xie, A multilevel correction method for Steklov eigenvalue problem by nonconforming finite element methods, Numer. Math. Theory Methods Appl., 8 (2015), 383-405.  doi: 10.4208/nmtma.2015.m1334.

[17]

J. R. Kuttler, Remarks on a Stekloff eigenvalue problem, SIAM J. Numer. Anal., 9 (1972), 1-5.  doi: 10.1137/0709001.

[18]

L. Li and J. An, An efficient spectral method and rigorous error analysis based on dimension reduction scheme for fourth order problems, Numer. Methods Partial Differential Equations, 37 (2021), 152-171.  doi: 10.1002/num.22523.

[19]

Q. LiQ. 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.

[20]

Q. Li and Y. Yang, A two-grid discretization scheme for the Steklov eigenvalue problem, J. Appl. Math. Comput., 36 (2011), 129-139.  doi: 10.1007/s12190-010-0392-9.

[21]

E. Sassone, Positivity for polyharmonic problems on domains close to a disk, Ann. Mat. Pura Appl., 186 (2007), 419-432.  doi: 10.1007/s10231-006-0012-3.

[22] J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006. 
[23]

J. Shen, T. Tang and L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 41. Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.

[24]

W. Stekloff, Sur les problèmes fondamentaux de la physique mathématique, Ann. Sci. École Norm. Sup., 19 (1902), 191-259.  doi: 10.24033/asens.510.

[25]

T. Tan and J. An, Spectral Galerkin approximation and rigorous error analysis for the Steklov eigenvalue problem in circular domain, Math. Methods Appl. Sci., 41 (2018), 3764-3778.  doi: 10.1002/mma.4863.

[26]

H. Xie, A type of multilevel method for the Steklov eigenvalue problem, IMA J. Numer. Anal., 34 (2014), 592-608.  doi: 10.1093/imanum/drt009.

[27]

F. Xu, A full multigrid method for the Steklov eigenvalue problem, Int. J. Comput. Math., 96 (2019), 2371-2386.  doi: 10.1080/00207160.2018.1562060.

[28]

F. XuL. Chen and Q. Huang, Local defect-correction method based on multilevel discretization for Steklov eigenvalue problem, ESAIM Math. Model. Numer. Anal., 55 (2021), 2899-2920.  doi: 10.1051/m2an/2021076.

[29]

Y. YangQ. 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.

show all references

References:
[1]

J. AnH. Bi and Z. Luo, A highly efficient spectral-Galerkin method based on tensor product for fourth-order Steklov equation with boundary eigenvalue, J. Inequal. Appl., 2016 (2016), 1-12.  doi: 10.1186/s13660-016-1158-1.

[2]

J. AnH. Li and Z. Zhang, Spectral-Galerkin approximation and optimal error estimate for biharmonic eigenvalue problems in circular/spherical/elliptical domains, Numer. Algorithms, 84 (2020), 427-455.  doi: 10.1007/s11075-019-00760-4.

[3]

J. An and Z. Zhang, An efficient spectral-Galerkin approximation and error analysis for Maxwell transmission eigenvalue problems in spherical geometries, J. Sci. Comput., 75 (2018), 157-181.  doi: 10.1007/s10915-017-0528-2.

[4]

A. B. Andreev and T. D. 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.

[5] S. Bergman and M. Schiffer, Kernel Functions and Elliptic Differential Equations in Mathematical Physics, Academic Press, Inc., New York, N.Y., 1953. 
[6]

E. BerchioF. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems, J. Differential Equations, 229 (2006), 1-23.  doi: 10.1016/j.jde.2006.04.003.

[7]

A. BermúdezR. Rodríguez 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.

[8]

H. BiH. Li and Y. Yang, An adaptive algorithm based on the shifted inverse iteration for the Steklov eigenvalue problem, Appl. Numer. Math., 105 (2016), 64-81.  doi: 10.1016/j.apnum.2016.02.003.

[9]

H. Bi, S. Ren and Y. Yang, Conforming finite element approximations for a fourth-order Steklov eigenvalue problem, Math. Probl. Eng., 2011 (2011), Art. ID 873152, 13 pp. doi: 10.1155/2011/873152.

[10]

H. Bi and Y. Yang, A two-grid method of the non-conforming crouzeix–raviart element for the Steklov eigenvalue problem, Appl. Math. Comput., 217 (2011), 9669-9678.  doi: 10.1016/j.amc.2011.04.051.

[11]

D. BucurA. Ferrero and F. Gazzola, On the first eigenvalue of a fourth order Steklov problem, Calc. Var. Partial Differential Equations, 35 (2009), 103-131.  doi: 10.1007/s00526-008-0199-9.

[12]

D. Bucur and F. Gazzola, The first biharmonic Steklov eigenvalue: Positivity preserving and shape optimization, Milan J. Math., 79 (2011), 247-258.  doi: 10.1007/s00032-011-0143-x.

[13]

C. Conca, J. Planchard and M. Vanninathan, Fluids and Periodic Structures, John Wiley and Sons, Ltd., Chichester; Masson, Paris, 1995.

[14]

A. FerreroF. Gazzola and T. Weth, On a fourth order Steklov eigenvalue problem, Analysis, 25 (2005), 315-332.  doi: 10.1524/anly.2005.25.4.315.

[15]

F. Gazzola and G. Sweers, On positivity for the biharmonic operator under Steklov boundary conditions, Arch. Ration. Mech. Anal., 188 (2008), 399-427.  doi: 10.1007/s00205-007-0090-4.

[16]

X. HanY. Li and H. Xie, A multilevel correction method for Steklov eigenvalue problem by nonconforming finite element methods, Numer. Math. Theory Methods Appl., 8 (2015), 383-405.  doi: 10.4208/nmtma.2015.m1334.

[17]

J. R. Kuttler, Remarks on a Stekloff eigenvalue problem, SIAM J. Numer. Anal., 9 (1972), 1-5.  doi: 10.1137/0709001.

[18]

L. Li and J. An, An efficient spectral method and rigorous error analysis based on dimension reduction scheme for fourth order problems, Numer. Methods Partial Differential Equations, 37 (2021), 152-171.  doi: 10.1002/num.22523.

[19]

Q. LiQ. 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.

[20]

Q. Li and Y. Yang, A two-grid discretization scheme for the Steklov eigenvalue problem, J. Appl. Math. Comput., 36 (2011), 129-139.  doi: 10.1007/s12190-010-0392-9.

[21]

E. Sassone, Positivity for polyharmonic problems on domains close to a disk, Ann. Mat. Pura Appl., 186 (2007), 419-432.  doi: 10.1007/s10231-006-0012-3.

[22] J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006. 
[23]

J. Shen, T. Tang and L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 41. Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.

[24]

W. Stekloff, Sur les problèmes fondamentaux de la physique mathématique, Ann. Sci. École Norm. Sup., 19 (1902), 191-259.  doi: 10.24033/asens.510.

[25]

T. Tan and J. An, Spectral Galerkin approximation and rigorous error analysis for the Steklov eigenvalue problem in circular domain, Math. Methods Appl. Sci., 41 (2018), 3764-3778.  doi: 10.1002/mma.4863.

[26]

H. Xie, A type of multilevel method for the Steklov eigenvalue problem, IMA J. Numer. Anal., 34 (2014), 592-608.  doi: 10.1093/imanum/drt009.

[27]

F. Xu, A full multigrid method for the Steklov eigenvalue problem, Int. J. Comput. Math., 96 (2019), 2371-2386.  doi: 10.1080/00207160.2018.1562060.

[28]

F. XuL. Chen and Q. Huang, Local defect-correction method based on multilevel discretization for Steklov eigenvalue problem, ESAIM Math. Model. Numer. Anal., 55 (2021), 2899-2920.  doi: 10.1051/m2an/2021076.

[29]

Y. YangQ. 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.

Figure 1.  The comparison between approximation solutions $ v_{MN} $ for $ N = 30,M = 15 $(left), $ N = 50, M = 25 $(center), and reference solution(right)
Figure 2.  The error between reference solution and approximation solutions $ v_{MN} $ for $ N = 30,M = 15 $(left), $ N = 50 $ and $ M = 25 $(right)
Figure 3.  The error tendency for the fixed $ M $(left) and the fixed $ N $(right)
Figure 4.  The error between exact solution and approximation solution for $ N = 30 $(left), $ N = 50 $(right)
Table 1.  The errors E(v, vMN) with different N and M.
N M=5 M=10 M=15 M=20 M=25
10 9.6887e-07 1.0025e-04 1.0637e-04 1.0676e-04 1.0685e-04
20 5.4884e-08 5.6948e-12 4.7867e-08 9.1367e-06 1.0056e-05
30 5.4884e-08 4.7638e-13 1.2520e-17 7.9545e-13 9.1848e-09
40 5.4884e-08 4.7638e-13 1.0309e-17 1.0308e-17 1.0367e-17
50 5.4884e-08 4.7638e-13 9.0532e-18 9.0513e-18 9.0513e-18
N M=5 M=10 M=15 M=20 M=25
10 9.6887e-07 1.0025e-04 1.0637e-04 1.0676e-04 1.0685e-04
20 5.4884e-08 5.6948e-12 4.7867e-08 9.1367e-06 1.0056e-05
30 5.4884e-08 4.7638e-13 1.2520e-17 7.9545e-13 9.1848e-09
40 5.4884e-08 4.7638e-13 1.0309e-17 1.0308e-17 1.0367e-17
50 5.4884e-08 4.7638e-13 9.0532e-18 9.0513e-18 9.0513e-18
Table 2.  The first eigenvalue λmN for m = 0, 1, 2, 3 and different N.
N λ0N λ1N λ2N λ3N
4 1.999999999999998 4.000000000000001 5.999999999999999 8.059210526315789
6 1.999999999999999 3.999999999999998 5.999999999999998 7.999999999999998
8 2.000000000000000 4.000000000000001 6.000000000000001 8.000000000000004
10 2.000000000000001 4.000000000000001 6.000000000000003 7.999999999999993
12 2.000000000000001 4.000000000000001 6.000000000000002 8.000000000000007
N λ0N λ1N λ2N λ3N
4 1.999999999999998 4.000000000000001 5.999999999999999 8.059210526315789
6 1.999999999999999 3.999999999999998 5.999999999999998 7.999999999999998
8 2.000000000000000 4.000000000000001 6.000000000000001 8.000000000000004
10 2.000000000000001 4.000000000000001 6.000000000000003 7.999999999999993
12 2.000000000000001 4.000000000000001 6.000000000000002 8.000000000000007
Table 3.  The first eigenvalue λmN for m = 0, 1, 2, 3 and different h.
h λ0h λ1h λ2h λ3h
1/4 2.000000032363347 4.000000018720217 6.000436469604340 8.005030682333810
1/6 2.000000028811607 4.000000048950857 6.000081442394085 8.000975861078535
1/8 2.000000024461073 4.000000073005518 6.000025208583129 8.000307091454056
1/10 2.000000022604670 4.000000056975401 6.000010214870977 8.000125489065235
1/12 2.000000020581500 4.000000061763554 6.000004896294027 8.000060442986863
h λ0h λ1h λ2h λ3h
1/4 2.000000032363347 4.000000018720217 6.000436469604340 8.005030682333810
1/6 2.000000028811607 4.000000048950857 6.000081442394085 8.000975861078535
1/8 2.000000024461073 4.000000073005518 6.000025208583129 8.000307091454056
1/10 2.000000022604670 4.000000056975401 6.000010214870977 8.000125489065235
1/12 2.000000020581500 4.000000061763554 6.000004896294027 8.000060442986863
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