Spectral-Galerkin approximation and optimal error estimate for biharmonic eigenvalue problems in circular/spherical/elliptical domains
Spectral-Galerkin approximation and optimal error estimate for biharmonic eigenvalue problems in circular/spherical/elliptical domains
复制标题
圆/球/椭圆域双调和特征值问题的谱伽辽金近似和最优误差估计
DOI：
10.1007/s11075-019-00760-4
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发表时间：
2019
影响因子：
2.1
通讯作者：
Zhimin Zhang
中科院分区：
文献类型：
--
作者：
Jing An;Huiyuan Li;Zhimin Zhang
本文提出并分析了圆/球/椭圆域双调和特征值问题的谱伽辽金方法。首先分析了极坐标下的四阶特征函数方程，然后导出了极点条件，并将圆盘/球体上的问题简化为一系列等价的一维特征值问题，这些问题可以并行求解。该方法的新颖之处在于根据极点条件构造适当加权的Sobolev空间，在此基础上，可以得到每个一维问题的近似特征值的最优误差估计。进一步，我们将该方法推广到椭圆域上的不可分双调和特征值问题，并建立了最优误差界。最后，我们提供了一些数值实验来验证我们的理论结果和算法。
In this paper, we propose and analyze spectral-Galerkin methods for the biharmonic eigenvalue problem in circular/spherical/elliptical domains. We first analyze the eigenfunction formulated fourth-order equation under the polar coordinates, then we derive the pole condition and reduce the problem on a circular disk/sphere to a sequence of equivalent one-dimensional eigenvalue problems that can be solved in parallel. The novelty of our approach lies in the construction of suitably weighted Sobolev spaces according to the pole conditions, based on which, the optimal error estimate for approximated eigenvalue of each one-dimensional problem can be obtained. Further, we extend our method to the non-separable biharmonic eigenvalue problem in an elliptic domain and establish the optimal error bounds. Finally, we provide some numerical experiments to validate our theoretical results and algorithms.