球域上椭圆方程有效的有限元方法和误差分析
An Effective Finite Element Method and Error Analysis for the Elliptic Equation in Spherical Domain
摘要: 本文提出了一种有效的有限元方法来求解球域上的椭圆方程。首先，利用球坐标变换和球谐函数展开，将原问题化为一系列等价的一维问题。其次，通过引入带权的Sobolev空间，建立了每个一维问题的弱形式和相应的离散形式。另外，利用分片线性插值多项式的逼近性质证明了逼近解的误差估计。最后给出了算法的具体实现过程，并通过数值例子验证了算法的有效性。
Abstract: In this paper, an effective finite element method is proposed to solve the elliptic equation in spherical domain. Firstly, a series of one-dimensional problems equivalent to the original problem are derived by introducing spherical coordinate transformation and using spherical harmonic functions expansion. Secondly, we introduce a weighted Sobolev space and establish the weak form and the corresponding discrete scheme. In addition, by using the approximation properties of piecewise linear interpolation polynomials, the error estimates of approximation solutions are proved. Finally, the concrete process of implementing the algorithm is given, and the effectiveness of the algorithm is verified by numerical experiments.
文章引用：彭娜. 球域上椭圆方程有效的有限元方法和误差分析[J]. 应用数学进展, 2021, 10(1): 66-73. https://doi.org/10.12677/AAM.2021.101008

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