Improved Numerical Quadrature over General Two-Dimensional Domains Utilizing Curvature Based Domain Decomposition

Herein we improve upon QUAD2D, an algorithm for numerical quadrature on a general two-dimensional domain presented by James Lambers. QUAD2D eliminates the error of approximation inherent in general polygonal methods due to the variance between an estimated polygon boundary and the domain boundary. The QUAD2D method achieves quadrature accurate to within machine precision for sufficiently smooth integrands, but has two weaknesses which are addressed in our proposed curvature based algorithm. The first weakness is that the boundary node location method utilizes a one-dimensional change of variable, causing it to be dependent on domain orientation. The second is that the domain decomposition results in some regions along the domain boundary edge including points outside the domain. Subsequent integration over these regions evaluates the integrand at points outside the domain where it might not even be defined. In regions where this occurs a correction is required. Our Curvature based Method (CBM) makes improvements to the QUAD2D algorithm addressing these weaknesses. The CBM boundary node location method utilizes a two-dimensional change of variable enabling the algorithm to be independent of domain orientation. The domain decomposition method also ensures that all regions to be integrated are inside the domain