Implementation of Gaussian Quadrature for Double Integrals over Arbitrary Planar Regions

Gaussian quadrature is an efficient numerical integration method particularly well-suited for computing double integrals over irregular planar regions. This approach achieves high accuracy with relatively few computational points by selecting optimal integration nodes and corresponding weights. In code implementation, this typically involves creating arrays for node coordinates and weight values, then computing weighted sums of function evaluations.

Core Implementation Strategy Region Processing: The integration domain is first mapped to a standard reference region (such as a square or triangle) through coordinate transformation. In programming terms, this involves implementing transformation functions that convert coordinates between the physical domain and reference domain using Jacobian determinants for area scaling. Node Selection: The method utilizes Gaussian quadrature nodes (like Legendre-Gauss nodes) for discretization. Programmatically, these nodes are typically precomputed using orthogonal polynomial roots and stored in lookup tables or calculated using specialized algorithms like the Golub-Welsch method. Weight Calculation: Each node carries a specific weight value, and the integral is approximated through weighted summation. Code implementation involves multiplying function values at nodes by their corresponding weights and summing the results, often achieved through vectorized operations for efficiency. Parameter Support: When integrand functions contain additional parameters, these can be directly embedded during the integration process without affecting numerical stability. This is implemented by designing functions that accept parameters as arguments and properly handling them during evaluation.

Advantages High Accuracy: Compared to trapezoidal or Simpson's methods, Gaussian quadrature achieves equal or better precision with fewer nodes. This translates to reduced computational complexity in code implementations. Flexibility: Suitable for complex boundary regions as long as appropriate coordinate transformations can be defined. This requires robust mesh generation and transformation algorithms in practical implementations. Computational Efficiency: Minimizes redundant calculations, making it particularly suitable for scenarios requiring repeated integration, such as optimization loops or parametric studies.

This method finds widespread application in engineering computations and scientific simulations, including electromagnetic field analysis, fluid dynamics modeling, and finite element methods where efficient numerical integration is crucial for performance.