求解流固振动Laplace模型的基于Rayleigh商移位反迭代的多网格方案
Multiscale Discretization Scheme Based on the Rayleigh Quotient Inverse Iteration Method for the Laplace Model for Fluid-Solid Vibrations
DOI: 10.12677/AAM.2021.106232, PDF,    国家自然科学基金支持
作者: 杜小虎, 闭 海：贵州师范大学数学科学学院，贵州 贵阳
关键词: 流固振动模型；有限元；多网格方案；Rayleigh商迭代；误差估计；Fluid-Structure； Finite Element； Multi-Scale Discretization； Rayleigh Quotient Inverse Iteration； Error Estimate
摘要: 本文讨论流固振动Laplace模型，首先建立该问题的基于Rayleigh商移位反迭代的多网格离散方案，利用该方案将在细网格上求解特征值问题归结为在粗网格上解特征值问题和在细网格上解一系列线性代数系统。其次分析多网格离散方案的误差。最后给出数值算例验证离散方案的有效性。
Abstract: In this paper, for the Laplace model for fluid-solid vibrations we establish the multiscale discretization scheme based on the Rayleigh quotient inverse iteration method. With this scheme, the solution of an eigenvalue problem on fine meshes is reduced to the solution of the eigenvalue problem on coarse meshes and the solutions of linear algebraic systems. Then we analyze and give the error estimates of the proposed scheme. Finally we present numerical experiments to validate the efficiency of the scheme.
文章引用：杜小虎, 闭海. 求解流固振动Laplace模型的基于Rayleigh商移位反迭代的多网格方案[J]. 应用数学进展, 2021, 10(6): 2226-2239. https://doi.org/10.12677/AAM.2021.106232

参考文献

[1] Xu, J. (1992) A New Class of Iterative Methods for Nonselfadjoint or Indefinite Problems. SIAM Journal on Numerical Analysis, 29, 303-319. [Google Scholar] [CrossRef] 
[2] Xu, J. (1996) Two-Grid Discretization Techniques for Linear and Nonlinear PDEs. SIAM Journal on Numerical Analysis, 33, 1759-1777. [Google Scholar] [CrossRef] 
[3] Xu, J. and Zhou, A. (2001) Two-Grid Discretization Scheme for Eigenvalue Problems. Mathematics of Computation, 70, 17-25. [Google Scholar] [CrossRef] 
[4] Racheva, M.R. and Andreev, A.B. (2002) Super Convergence Postprocessing for Eigenvalues. Computational Methods in Applied Mathematics, 2, 171-185. [Google Scholar] [CrossRef] 
[5] Chien, C.S. and Jeng, B.W. (2006) A Two-Grid Discretization Scheme for Semilinear Elliptic Eigenvalue Problems. SIAM Journal on Scientific Computing, 27, 1287-1304. [Google Scholar] [CrossRef] 
[6] Dai, X. and Zhou, A. (2008) Three-Scale Finite Element Discretizations for Quantum Eigenvalue Problems. SIAM Journal on Numerical Analysis, 46, 295-324. [Google Scholar] [CrossRef] 
[7] Chen, H., Jia, S. and Xie, H. (2009) Postprocessing and Higher Order Convergence for the Mixed Finite Element Approximations of the Stokes Eigenvalue Problems. Applications of Mathematics, 54, 237-250. [Google Scholar] [CrossRef] 
[8] Xie, H. and Yin, X. (2015) Acceleration of Stabilized Finite Element Discretizations for the Stokes Eigenvalue Problem. Advances in Computational Mathematics, 41, 799-812. [Google Scholar] [CrossRef] 
[9] Chen, J., Xu, Y. and Zou, J. (2010) An Adaptive Inverse Iteration for Maxwell Eigenvalue Problem Based on Edge Elements. Journal of Computational Physics, 229, 2649-2658. [Google Scholar] [CrossRef] 
[10] Andreev, A., Lazarov, R. and Racheva, M. (2005) Postprocessing and Higher Order Convergence of the Mixed Finite Element Approximations of Biharmonic Eigenvalue Problems. Journal of Computational and Applied Mathematics, 182, 333-349. [Google Scholar] [CrossRef] 
[11] Hu, X. and Cheng, X. (2011) Acceleration of a Two-Grid Method for Eigenvalue Problems. Mathematics of Computation, 80, 1287-1301. [Google Scholar] [CrossRef] 
[12] Hu, X. and Cheng, X. (2015) Corrigendum to: Acceleration of a Two-Grid Method for Eigenvalue Problems. Mathematics of Computation, 84, 2701-2704. [Google Scholar] [CrossRef] 
[13] Yang, Y. and Bi, H. (2011) Two-Grid Finite Element Discretization Scheme Based on Shifted-Inverse Power Method for Elliptic Eigenvalue Problems. SIAM Journal on Numerical Analysis, 49, 1602-1624. [Google Scholar] [CrossRef] 
[14] Chen, H., He, Y., Li, Y. and Xie, H. (2015) A Multigrid Method for Eigenvalue Problems Based on Shifted-Inverse Power Technique. European Journal of Mathematics, 1, 207-228. [Google Scholar] [CrossRef] 
[15] Zhou, J., Hu, X., Shu, S., Zhong, L. and Chen, L. (2014) Two-Grid Methods for Maxwell Eigenvalue Problems. SIAM Journal on Numerical Analysis, 52, 2027-2047. [Google Scholar] [CrossRef] [PubMed]
[16] Liu, J., Jiang, W., Lin, F., Liu, N. and Liu, Q. (2017) A Two-Grid Vector Discretization Scheme for the Resonant Cavity Problem with Anisotropic Media. IEEE Transactions on Microwave Theory and Techniques, 65, 2719-2725. [Google Scholar] [CrossRef] 
[17] Han, J., Zhang, Z. and Yang, Y. (2015) A New Adaptive Mixed Finite Element Method Based on Residual Type a Posterior Error Estimates for the Stokes Eigenvalue Problem. Numerical Methods for Partial Differential Equations, 31, 31-53. [Google Scholar] [CrossRef] 
[18] Bi, H. and Yang, Y. (2012) Multiscale Discretization Scheme Based on the Rayleigh Quotient Iterative Method for the Steklov Eigenvalue Problem. Mathematical Problems in Engineering, 2012, Article ID: 487207. [Google Scholar] [CrossRef] 
[19] Yang, Y., Bi, H., Han, J. and Yu, Y. (2015) The Shifted-Inverse Iteration Based on the Multigrid Discretizations for Eigenvalue Problems. SIAM Journal on Scientific Computing, 37, A2583-A2606. [Google Scholar] [CrossRef] 
[20] Zhang, Y., Bi, H. and Yang, Y. (2019) The Two-Grid Discretization of Ciarlet-Raviart Mixed Method for Biharmonic Eigenvalue Problems. Applied Numerical Mathematics, 138, 94-113. [Google Scholar] [CrossRef] 
[21] Lin, Q. and Xie, H. (2015) A Multi-Level Correction Scheme for Eigenvalue Problems. Mathematics of Computation, 84, 71-88. [Google Scholar] [CrossRef] 
[22] Conca, C., Osses, A. and Planchard, J. (1998) Asymptotic Analysis Relating Spectral Models in Fluid-Solid Vibrations. SIAM Journal on Numerical Analysis, 35, 1020-1048. [Google Scholar] [CrossRef] 
[23] Planchard, J. (1983) Eigen Frequencies of a Tube Bundle Placed in a Confined Fluid. Computer Methods in Applied Mechanics and Engineering, 30, 75-93. [Google Scholar] [CrossRef] 
[24] Planchard, J. and Ibnou-Zahir, M. (1983) Natural Frequencies of Tube Bundle in an Uncompressible Fluid. Computer Methods in Applied Mechanics and Engineering, 41, 47-68. [Google Scholar] [CrossRef] 
[25] Armentano, M.G., Padra, C., Rodríguez, R. and Scheble, M. (2011) An hp Finite Element Adaptive Scheme to Solve the Laplace Model for Fluid-Solid Vibrations. Computer Methods in Applied Mechanics and Engineering, 200, 178-188. [Google Scholar] [CrossRef] 
[26] Zhang, Y. and Yang, Y. (2021) Guaranteed Lower Eigenvalue Bounds for Two Spectral Problems Arising in Fluid Mechanics. Computers & Mathematics with Applications, 90, 66-72. [Google Scholar] [CrossRef] 
[27] Conca, C., Planchard, J. and Vanninathan, M. (1995) Fluid and Periodic Structrues. Masson, Paris.
[28] Grisvard, P. (1985) Elliptic Problems in Nonsmooth Domain. Pitman, Boston.
[29] Bramble, J.H. and Osborn, J.E. (1972) Approximation of Steklov Eigenvalues of Non-Selfadjoint Second Order Elliptic Operators. Academic, New York, 387-408. [Google Scholar] [CrossRef] 
[30] Babuška, I. and Osborn, J. (1991) Eigenvalue Problems. In: Ciarlet, P.G. and Lions, J.L., Eds., Finite Element Methods (Part 1), Handbook of Numerical Analysis, Vol. 2, Elsevier Science Publishers, North-Holland, 641-787. [Google Scholar] [CrossRef] 
[31] Chen, L. (2009) iFEM: An Integrated Finite Element Method Package in MATLAB. Technical Report, University of California at Irvine, Irvine.

为你推荐