双调和算子特征值问题的混合三角谱元方法

doi:10.12286/jssx.2017.1.81

计算数学 ›› 2017, Vol. 39 ›› Issue (1): 81-97.DOI: 10.12286/jssx.2017.1.81

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. 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.

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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 Weikun^1,2, Li Huiyuan³

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.

关键词： 三角谱元, 混合方法, 双调和算子特征值, 广义Koornwinder多项式, 误差估计

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.

Key words: triangular spectral element, mixed methods, biharmonic eigenvalues, generalized Koornwinder polynomials, error estimates

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摘要

双调和算子特征值问题的混合三角谱元方法

A MIXED TRIANGULAR SPECTRAL ELEMENT METHOD OF BIHARMONIC EIGENVALUE PROBLEM

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