Advanced Methodological Framework of Nmm Analysis: Formulation, Integration, and Solution Strategies for the Laplace Equation Problem with Complex Boundaries

Abstract

Numerical computation of partial differential equations (PDEs) is of high importance across numerous scientific and engineering disciplines. The accuracy and convergence of integration-based methods primarily depend on the capability to perform analytical integrations over complex domains. Despite the inherent challenges posed by the complexities of irregular integration domains and general integrands, which often render numerical integration impractical or even infeasible, this paper introduces an innovative analytical method for non-polynomial integration over complex domains. This method is initially applied within the Numerical Manifold Method (NMM) to address the inevitable trigonometric and exponential polynomial integrations encountered in the analysis of the Laplace equation. First, this paper provides a comprehensive overview of the fundamentals of NMM and the Simplex Integration (SI) method. Subsequently, it elaborates on the NMM framework for solving the Laplace equation, with a specific emphasis on deriving closed-form formulas for the pertinent trigonometric and exponential polynomial integration. Finally, some rigorous numerical experiments are conducted to validate the feasibility and practicality of the proposed method. In conclusion, this study enhances NMM by introducing an innovative analytical integration method for non-polynomial integration over complex domains; furthermore, it promises to significantly enhance the accuracy and convergence across a wide range of integration-based methods.

Keywords: Partial differential equation, Numerical manifold method, Non-polynomial simplex integration, Trigonometric and exponential polynomials, Laplace equation