Numerical Integration in Meshless Galerkin Methods with Multi-Point Gaussian Quadrature

In meshless Galerkin methods, numerical integration serves as a crucial computational component. This program provides a comprehensive implementation featuring multi-point Gaussian quadrature, allowing users to select integration points based on specific requirements. The implementation supports adaptive integration schemes where users can specify the number of Gaussian points (typically 2, 4, 6, or more) depending on the required precision level and problem complexity. The Gaussian quadrature algorithm employed in this code efficiently calculates integrals by optimizing the placement of integration points and corresponding weights, following the mathematical foundation of orthogonal polynomials. This method demonstrates exceptional computational efficiency for various numerical computation problems, particularly in handling complex domain integrations common in meshless methods. Key features include: - Customizable integration point selection through parameter configuration - Implementation of standard Gaussian quadrature formulas with precision up to 2n-1 for n points - Efficient handling of complex mathematical operations including polynomial integrations Using multi-point Gaussian integration significantly enhances computational accuracy and provides robust solutions for challenging problems such as partial differential equation solving. The program incorporates error control mechanisms and convergence validation to ensure reliable results. This implementation substantially improves user productivity by offering a versatile integration tool that effectively addresses practical engineering and scientific computing challenges.