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双调和算子特征值问题的混合三角谱元方法

单炜琨1,2, 李会元3   

  1. 1. 中国科学院软件研究所, 北京 100190;
    2. 中国科学院大学, 北京 100190;
    3. 中国科学院软件研究所, 北京 100190
  • 收稿日期:2016-05-03 出版日期:2017-02-15 发布日期:2017-02-17
  • 基金资助:

    本研究课题受国家自然科学基金(No.91130014,No.11471312,No.91430216)资助.

单炜琨, 李会元. 双调和算子特征值问题的混合三角谱元方法[J]. 计算数学, 2017, 39(1): 81-97.

Shan Weikun, Li Huiyuan. A MIXED TRIANGULAR SPECTRAL ELEMENT METHOD OF BIHARMONIC EIGENVALUE PROBLEM[J]. Mathematica Numerica Sinica, 2017, 39(1): 81-97.

A MIXED TRIANGULAR SPECTRAL ELEMENT METHOD OF BIHARMONIC EIGENVALUE PROBLEM

Shan Weikun1,2, Li Huiyuan3   

  1. 1. Institute of Software, Chinese Academy of Sciences, Beijing 100190, China;
    2. University of Chinese Academy of Sciences, Beijing 100190, China;
    3. Institute of Software, Chinese Academy of Sciences, Beijing 100190, China
  • Received:2016-05-03 Online:2017-02-15 Published:2017-02-17
本文针对双调和算子特征值问题设计了基于混合变分形式的三角谱元逼近格式,其基函数采用指标为(-1,-1,-1)的广义Koornwinder多项式.在H1-及H01-正交谱元投影的逼近理论基础上,我们建立了双调和算子特征值与特征函数的收敛性估计;它关于网格尺寸h是最优的,关于多项式次数M是次优的.然而,在H02-正交谱元投影的最优估计假设前提下,关于M的次优收敛阶估计则提升为最优.此外,Koornwinder分片多项式逼近的结果还表明,在带权Besov空间范数的度量下,对于存在着区域角点奇性的双调和算子特征值问题,谱元方法的收敛阶能达到h-型有限元方法的2倍.最后,本文的数值实验结果展示了谱元逼近格式的高效性,同时也验证了相关理论的正确性.
A triangular spectral element approximation scheme using generalized Koornwinder polynomials of index (-1,-1,-1) is proposed and analyzed for the biharmonic eigenvalue problem based on its mixed variational formulation.Further,on the basis of approximation theories of the H1-and H01-orthogonal spectral element projections oriented to the secondorder equations,error estimates are eventually established for our mixed triangular spectral element method (TSEM),which are optimal with respect to the mesh size h and sub-optimal with respected to the polynomial degree M.However,under certain assumption for the H02-orthogonal spectral element projection,we can also obtain an optimal estimate with respect to M.The approximation results of piecewise Koornwinder polynomials show that,under the measurement in some weighted Besov spaces,TSEM converges twice as fast as the hversion finite element method if the eigenfunction of the biharmonic operator has corner singularity.Finally,numerical results show the effectivity of our mixed TSEM and illustrate our theories as well.

MR(2010)主题分类: 

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[1] Brezzi F, Fortin M. Mixed and hybrid finite element methods[M]. Springer Science & Business Media, 2012.

[2] Scholtz R. A mixed method for fourth-order problems using the linear finite elements[J]. RAIRO Numer. Anal, 1978, 15:85-90.

[3] Roberts J E, Thomas J M. Mixed and hybrid methods[J]. Handbook of numerical analysis, 1991, 2:523-639.

[4] Bernardi C, Maday Y. Spectral methods[J]. Handbook of numerical analysis, 1997, 5:209-485.

[5] Bernardi C, Coppoletta G, Maday Y. Some spectral approximations of two-dimensional fourthorder problems[J]. Mathematics of computation, 1992, 59(199):63-76.

[6] Koornwinder T. Two-variable analogues of the classical orthogonal polynomials[J]. Theory and applications of special functions, 1975:435-495.

[7] Guo B Y, Shen J, Wang L L. Generalized Jacobi polynomials/functions and their applications[J]. Applied Numerical Mathematics, 2009, 59(5):1011-1028.

[8] Mercier B, Osborn J, Rappaz J, et al. Eigenvalue approximation by mixed and hybrid methods[J]. Mathematics of Computation, 1981, 36(154):427-453.

[9] Polizzi E. Density-matrix-based algorithm for solving eigenvalue problems[J]. Physical Review B, 2009, 79(11):115112.

[10] Guo B, Wang L L. Error analysis of spectral method on a triangle[J]. Advances in Computational Mathematics, 2007, 26(4):473-496.

[11] Li H, Shen J. Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle[J]. Mathematics of Computation, 2010, 79(271):1621-1646.

[12] Schwab C. p-and hp-finite element methods:Theory and applications in solid and fluid mechanics[M]. Oxford University Press, 1998.

[13] Canuto C G, Hussaini M Y, Quarteroni A M, et al. Spectral Methods:Evolution to Complex Geometries and Applications to Fluid Dynamics (Scientific Computation)[M]. Springer-Verlag New York, Inc., 2007.

[14] Weikun Shan and Huiyuan Li. The triangular spectral element method for Stokes eigenvalues[J]. Math-ematics of Computation, 2016, Accepted.

[15] Maz'ya V. Sobolev Spaces:with Applications to Elliptic Partial Differential Equations[J]. 2011.

[16] Babuska I, Guo B. Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces. part I:Approximability of functions in the weighted Besov spaces[J]. SIAM Journal on Numerical Analysis, 2002, 39(5):1512-1538.

[17] Falk R S, Osborn J E. Error estimates for mixed methods[J]. RAIRO-Analyse num W rique, 1980, 14(3):249-277.

[18] Boffi D, Brezzi F, Gastaldi L. On the convergence of eigenvalues for mixed formulations[J]. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 1997, 25(1-2):131-154.

[19] Blum H, Rannacher R, Leis R. On the boundary value problem of the biharmonic operator on domains with angular corners[J]. Mathematical Methods in the Applied Sciences, 1980, 2(4):556-581.

[20] Grisvard P. Elliptic problems in nonsmooth domains[M]. SIAM, 2011.

[21] Gerasimov T, Stylianou A, Sweers G. Corners give problems when decoupling fourth order equations into second order systems[J]. SIAM Journal on Numerical Analysis, 2012, 50(3):1604-1623.

[22] Babuška I, Guo B Q. Approximation properties of the h-p version of the finite element method[J]. Computer Methods in Applied Mechanics and Engineering, 1996, 133(3):319-346.
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