An efficient Fourier spectral method and error analysis for the fourth order problem with periodic boundary conditions and variable coefficients

Tingting Jiang; Jiantao Jiang; Jing An; Tingting Jiang; Jiantao Jiang; Jing An

doi:10.3934/math.2023484

AIMS Mathematics

2023, Volume 8, Issue 4: 9585-9601. doi: 10.3934/math.2023484

Previous Article Next Article

Research article

An efficient Fourier spectral method and error analysis for the fourth order problem with periodic boundary conditions and variable coefficients

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

Received: 05 December 2022 Revised: 01 February 2023 Accepted: 08 February 2023 Published: 20 February 2023
MSC : 65N15, 65N35

We propose in this paper an efficient algorithm based on the Fourier spectral-Galerkin approximation for the fourth-order elliptic equation with periodic boundary conditions and variable coefficients. First, by using the Lax-Milgram theorem, we prove the existence and uniqueness of weak solution and its approximate solution. Then we define a high-dimensional $ L^2 $ projection operator and prove its approximation properties. Combined with Céa lemma, we further prove the error estimate of the approximate solution. In addition, from the Fourier basis function expansion and the properties of the tensor, we establish the equivalent matrix form based on tensor product for the discrete scheme. Finally, some numerical experiments are carried out to demonstrate the efficiency of the algorithm and correctness of the theoretical analysis.
- fourth-order problems,
- periodic boundary conditions,
- variable coefficients,
- Fourier spectral method,
- error estimation
Citation: Tingting Jiang, Jiantao Jiang, Jing An. An efficient Fourier spectral method and error analysis for the fourth order problem with periodic boundary conditions and variable coefficients[J]. AIMS Mathematics, 2023, 8(4): 9585-9601. doi: 10.3934/math.2023484

Related Papers:

Abstract

We propose in this paper an efficient algorithm based on the Fourier spectral-Galerkin approximation for the fourth-order elliptic equation with periodic boundary conditions and variable coefficients. First, by using the Lax-Milgram theorem, we prove the existence and uniqueness of weak solution and its approximate solution. Then we define a high-dimensional $ L^2 $ projection operator and prove its approximation properties. Combined with Céa lemma, we further prove the error estimate of the approximate solution. In addition, from the Fourier basis function expansion and the properties of the tensor, we establish the equivalent matrix form based on tensor product for the discrete scheme. Finally, some numerical experiments are carried out to demonstrate the efficiency of the algorithm and correctness of the theoretical analysis.

References

[1]	D. F. Griffiths, A. R. Mitchell, J. L. Morris, A numerical study of the nonlinear Schrödinger equation, Comput. Method. Appl. Mech. Eng., 45 (1984), 177–215. https://doi.org/10.1016/0045-7825(84)90156-7 doi: 10.1016/0045-7825(84)90156-7
[2]	S. Momani, An explicit and numerical solutions of the fractional KdV equation, Math. Comput. Simul., 70 (2005), 110–118. https://doi.org/10.1016/j.matcom.2005.05.001 doi: 10.1016/j.matcom.2005.05.001
[3]	M. Boivin, O. Simonin, K. D. Squires, Direct numerical simulation of turbulence modulation by particles in isotropic turbulence, J. Fluid Mech., 375 (1998), 235–263. https://doi.org/10.1017/s0022112098002821 doi: 10.1017/s0022112098002821
[4]	J. Shen, Efficient spectral-Galerkin methods III: Polar and cylindrical geometries, SIAM J. Sci. Comput., 18 (1997), 1583–1604. https://doi.org/10.1137/S1064827595295301 doi: 10.1137/S1064827595295301
[5]	J. Shen, Efficient spectral-Galerkin methods IV. Spherical geometries, SIAM J. Sci. Comput., 20 (1999), 1438–1455. https://doi.org/10.1137/S1064827597317028 doi: 10.1137/S1064827597317028
[6]	G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. Mazzei, 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. Method. Appl. Mech. Eng., 191 (2002), 3669–3750. https://doi.org/10.1016/S0045-7825(02)00286-4 doi: 10.1016/S0045-7825(02)00286-4
[7]	B. Li, G. Fairweather, B. Bialecki, Discrete-time orthogonal spline collocation methods for vibration problems, SIAM J. Numer. Anal., 39 (2002), 2045–2065. https://doi.org/10.1137/S0036142900348729 doi: 10.1137/S0036142900348729
[8]	J. Shen, J. Xu, J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys., 353 (2018), 407–416. https://doi.org/10.1016/j.jcp.2017.10.021 doi: 10.1016/j.jcp.2017.10.021
[9]	Y. F. Wei, Q. L. Song, Z. B. Bai, Existence and iterative method for some fourth order nonlinear boundary value problems, Appl. Math. Lett., 87 (2019), 101–107. https://doi.org/10.1016/j.aml.2018.07.032 doi: 10.1016/j.aml.2018.07.032
[10]	J. Shen, J. Xu, J. Yang, A new class of efficient and robust energy stable schemes for gradient flows, SIAM Rev., 61 (2019), 474–506. https://doi.org/10.1137/17M1150153 doi: 10.1137/17M1150153
[11]	Z. H. Huo, Y. L. Jia, The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament, J. Differ. Equations, 214 (2005), 1–35. https://doi.org/10.1016/j.jde.2004.09.005 doi: 10.1016/j.jde.2004.09.005
[12]	S. Ren, T. Tan, J. An, An efficient spectral-Galerkin approximation based on dimension reduction scheme for transmission eigenvalues in polar geometries, Comput. Math. Appl., 80 (2020), 940–955. https://doi.org/10.1016/j.camwa.2020.05.018 doi: 10.1016/j.camwa.2020.05.018
[13]	T. Tan, W. Cao, J. An, Spectral approximation based on a mixed scheme and its error estimates for transmission eigenvalue problems, Comput. Math. Appl., 111 (2022), 20–33. https://doi.org/10.1016/J.CAMWA.2022.02.009 doi: 10.1016/J.CAMWA.2022.02.009
[14]	N. Peng, C. Wang, J. An, An efficient finite-element method and error analysis for the fourth-order elliptic equation in a circular domain, Int. J. Comput. Math., 99 (2022), 1785–1802. https://doi.org/10.1080/00207160.2021.2007240 doi: 10.1080/00207160.2021.2007240
[15]	N. Peng, J. Han, J. An, An efficient finite element method and error analysis for fourth order problems in a spherical domain, Discrete Cont. Dyn. B, 27 (2022), 6807–6821. https://doi.org/10.3934/dcdsb.2022021 doi: 10.3934/dcdsb.2022021
[16]	L. Ge, H. F. Niu, J. W. Zhou, Convergence analysis and error estimate for distributed optimal control problems. governed by Stokes equations with velocity-constraint, Adv. Appl. Math. Mech., 14 (2022), 33–55. https://doi.org/10.4208/aamm.OA-2020-0302 doi: 10.4208/aamm.OA-2020-0302
[17]	H. F. Niu, D. P. Yang, J. W. Zhou, Numerical analysis of an optimal control problem governed by the stationary Navier-Stokes equations with global velocity-constrained, Commun. Comput. Phys., 24 (2018), 1477–1502. https://doi.org/10.4208/cicp.oa-2017-0045 doi: 10.4208/cicp.oa-2017-0045
[18]	Y. P. Chen, J. W. Zhou, Error estimates of spectral Legendre-Galerkin methods for the fourth-order equation in one dimension, Comput. Math. Appl., 268 (2015), 1217–1226. https://doi.org/10.1016/j.amc.2015.06.082 doi: 10.1016/j.amc.2015.06.082
[19]	J. W. Zhou, J. Zhang, X. Q. Xing, Galerkin spectral approximations for optimal control problems governed by the fourth order equation with an integral constraint on state, Comput. Math. Appl., 72 (2016), 2549–2561. https://doi.org/10.1016/j.camwa.2016.08.009 doi: 10.1016/j.camwa.2016.08.009
[20]	B. Bialecki, A. Karageorghis, A Legendre spectral Galerkin method for the biharmonic Dirichlet problem, SIAM J. Sci. Comput., 22 (2001), 1549–1569. https://doi.org/10.1137/S1064827598342407 doi: 10.1137/S1064827598342407
[21]	P. E. BjØrstad, B. P. TjØstheim, Efficient algorithms for solving a fourth-order equation with spectral-Galerkin method, SIAM J. Sci. Comput., 18 (1997), 621–632. https://doi.org/10.1137/S1064827596298233 doi: 10.1137/S1064827596298233
[22]	E. H. Doha, A. H. Bhrawy, Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials, Appl. Numer. Math., 58 (2008), 1224–1244. https://doi.org/10.1016/j.apnum.2007.07.001 doi: 10.1016/j.apnum.2007.07.001
[23]	J. T. Jiang, J. An, J. W. Zhou, A novel numerical method based on a high order polynomial approximation of the fourth order Steklov equation and its eigenvalue problems, Discrete Cont. Dyn. B, 28 (2023), 50–69. https://doi.org/10.3934/dcdsb.2022066 doi: 10.3934/dcdsb.2022066
[24]	D. Q. Jiang, H. Z. Liu, L. L. Zhang, Optimal existence theory for single and multiple positive solutions to fourth-order periodic boundary value problems, Nonlinear Anal. Real World Appl., 7 (2006), 841–852. https://doi.org/10.1016/j.nonrwa.2005.05.003 doi: 10.1016/j.nonrwa.2005.05.003
[25]	J. Fialho, F. Minhós, Fourth order impulsive periodic boundary value problems, Differ. Equat. Dyn. Sys., 23 (2015), 117–127. https://doi.org/10.1007/s12591-013-0186-2 doi: 10.1007/s12591-013-0186-2
[26]	E. H. Doha, A. H. Bhrawy, Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials, Appl. Numer. Math., 58 (2008), 1224–1244. https://doi.org/10.1016/j.apnum.2007.07.001 doi: 10.1016/j.apnum.2007.07.001
[27]	Y. X. Li, Positive solutions of fourth-order boundary value problems with two parameters, J. Math. Anal. Appl., 281 (2003), 477–484. https://doi.org/10.1016/S0022-247X(03)00131-8 doi: 10.1016/S0022-247X(03)00131-8
[28]	A. R. Abdullaev, E. A. Skachkova, Periodic boundary-value problem for a fourth-order differential equation, Russ. Math., 57 (2013), 1–7. https://doi.org/10.3103/S1066369X13120013 doi: 10.3103/S1066369X13120013
[29]	S. B. Kuang, K. Li, R. P. Zou, R. H. Pan, A. B. Yu, Application of periodic boundary conditions to CFD-DEM simulation of gas-solid flow in pneumatic conveying, Chem. Eng. Sci., 93 (2013), 214–228. https://doi.org/10.1016/j.ces.2013.01.055 doi: 10.1016/j.ces.2013.01.055
[30]	M. Jbeili, J. F. Zhang, The generalized periodic boundary condition for microscopic simulations of heat transfer in heterogeneous materials, Int. J. Heat Mass Tran., 173 (2021). https://doi.org/10.1016/J.IJHEATMASSTRANSFER.2021.121200
[31]	Y. Liu, Z. Fang, H. Li, S. He, W. Gao, A coupling method based on new MFE and FE for fourth-order parabolic equation, J. Appl. Math. Comput., 43 (2013), 249–269. https://doi.org/10.1007/s12190-013-0662-4 doi: 10.1007/s12190-013-0662-4

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)