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Numerical Solution for Schrödinger Eigenvalue Problem Using Isogeometric Analysis on Implicit Domains
Communications in Mathematics and Statistics ( IF 0.9 ) Pub Date : 2019-07-26 , DOI: 10.1007/s40304-019-00186-3
Ammar Qarariyah , Fang Deng , Tianhui Yang , Jiansong Deng

We study the accuracy and performance of isogeometric analysis on implicit domains when solving time-independent Schrödinger equation. We construct weighted extended PHT-spline basis functions for analysis, and the domain is presented with same basis functions in implicit form excluding the need for a parameterization step. Moreover, an adaptive refinement process is formulated and discussed with details. The constructed basis functions with cubic polynomials and only \(C^{1}\) continuity are enough to produce a higher continuous field approximation while maintaining the computational cost for the matrices as low as possible. A numerical implementation for the adaptive method is performed on Schrödinger eigenvalue problem with double-well potential using 3 examples on different implicit domains. The convergence and performance results demonstrate the efficiency and accuracy of the approach.

中文翻译:

隐域等几何分析的Schrödinger特征值问题数值解

当求解与时间无关的Schrödinger方程时,我们研究了隐域上等几何分析的准确性和性能。我们构造了加权的扩展PHT样条基函数进行分析,并且以隐式形式为域提供了相同的基函数,而不需要参数化步骤。此外,制定并详细讨论了自适应细化过程。具有三次多项式且仅\(C ^ {1} \)的构造基函数连续性足以产生更高的连续场近似值,同时保持矩阵的计算成本尽可能低。通过在不同隐域上使用3个示例,对具有双阱势的Schrödinger特征值问题进行了自适应方法的数值实现。收敛性和性能结果证明了该方法的效率和准确性。
更新日期:2019-07-26
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