双调和算子特征值问题的混合三角谱元方法

发布时间：2018-06-08 13:43

本文选题：三角谱元 + 混合方法　；参考：《计算数学》2017年01期

【摘要】：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H~1-及H_0~1-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H_0~2-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.
[Abstract]:In this paper, a trigonometric spectral element approximation scheme based on mixed variational form is designed for biharmonic operator eigenvalue problems. The basis function of the scheme is generalized Koornwinder polynomials with the index of -1 ~ (-1) ~ (-1). On the basis of the approximation theory of H ~ 1- and H _ T _ 01- orthogonal spectral element projection, we establish the convergence estimates of the eigenvalues and eigenfunctions of biharmonic operators, which are optimal for mesh size h and suboptimal for polynomial degree M. However, under the assumption of optimal estimation of the projection of H _ 0S _ 2-orthogonal spectral element, the suboptimal convergence order estimate of M is promoted to the optimal one. In addition, the results of Koornwinder piecewise polynomial approximation show that under the metric of weighted Besov space norm, the convergence order of spectral element method can reach 2 times of that of h- type finite element method for biharmonic operator eigenvalue problems with regional singularity. Finally, the numerical results of this paper demonstrate the efficiency of spectral element approximation scheme, and verify the correctness of the related theory.
【作者单位】：中国科学院软件研究所;中国科学院大学;
【基金】：国家自然科学基金(No.91130014,No.11471312,No.91430216)资助
【分类号】：O241.82

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