A new method for numerical integration of singular functions on the plane

Abstract

Region transformation methods have been used by many authors to integrate singular multivariate integrands, the most notable being the Duffy transformation that maps squares onto triangles and cubes onto pyramids. In this paper, a new method that generalizes and improves the existing ones is provided. We show that behind these variable transformation methods there is an underlying linear structure involving the shape functions of the integration domains. This structure can be modified with further composition of the original transformation with regular mappings on the unit square in order to improve the performance of the numerical integration. The proposed method is neutral in the sense that the transformation composition is suitable for different types of singularities, without making any particular assumption on the regular part of the integrand. Moreover, this is achieved without a significant increase in the computational cost and making use of standard Gaussian quadrature rules only. We illustrate the efficiency of the proposed methods with comparative numerical examples. This technique proves to be both efficient and practical and has a better convergence than those based on classical transformations such as power or trigonometric maps.