Scholarlink

<p>The numerical computation of partial differential equations (PDEs) is highly important across numerous scientific and engineering disciplines. The accuracy and convergence of integration-based methods depend primarily on the ability to perform analytical integration over complex domains. Owing to the inherent challenges posed by the complexities of irregular integration domains and general integrands, this paper introduces an innovative analytical method for nonpolynomial integration over complex domains for the first time. This method is initially applied within the framework of the numerical manifold method (NMM) to address the inevitable trigonometric and exponential polynomial integrations encountered in the analysis of the Laplace equation problem. First, a comprehensive overview of the fundamentals of the NMM and the simplex integration (SI) method is provided in this paper. Subsequently, the NMM framework for solving the Laplace equation is elaborated upon, with a focus on deriving closed-form formulas for trigonometric and exponential polynomial integration. Finally, a series of rigorous numerical experiments is conducted, where the proposed method demonstrates improved accuracy and efficiency. In conclusion, this study innovatively enhances the NMM by introducing the SI method for nonpolynomial functions over complex domains, which is a promising approach for increasing accuracy and convergence across various integration-based methods. This groundbreaking achievement has not yet been reported in the publicly available literature.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Advanced methodological framework for NMM analysis: Formulation, integration, and solution strategies for the Laplace equation problem with complex boundaries

  • 期刊论文

  • 发表日期：

    2025年9月15日

查看全文
  • Xilong Li,
  • Hong Zhang,
  • Haocheng Huang,
  • Huanyan Lai,
  • Genhua Shi

摘要

The numerical computation of partial differential equations (PDEs) is highly important across numerous scientific and engineering disciplines. The accuracy and convergence of integration-based methods depend primarily on the ability to perform analytical integration over complex domains. Owing to the inherent challenges posed by the complexities of irregular integration domains and general integrands, this paper introduces an innovative analytical method for nonpolynomial integration over complex domains for the first time. This method is initially applied within the framework of the numerical manifold method (NMM) to address the inevitable trigonometric and exponential polynomial integrations encountered in the analysis of the Laplace equation problem. First, a comprehensive overview of the fundamentals of the NMM and the simplex integration (SI) method is provided in this paper. Subsequently, the NMM framework for solving the Laplace equation is elaborated upon, with a focus on deriving closed-form formulas for trigonometric and exponential polynomial integration. Finally, a series of rigorous numerical experiments is conducted, where the proposed method demonstrates improved accuracy and efficiency. In conclusion, this study innovatively enhances the NMM by introducing the SI method for nonpolynomial functions over complex domains, which is a promising approach for increasing accuracy and convergence across various integration-based methods. This groundbreaking achievement has not yet been reported in the publicly available literature.