Generalized Gaussian quadrature rules over two-dimensional regions with linear sides

Abstract

This paper presents a generalized Gaussian quadrature method for numerical integration over triangular, parallelogram and quadrilateral elements with linear sides. In order to derive the quadrature rule, a general transformation of the regions, R1={(x,y)∣a⩽x⩽b,g(x)⩽y⩽h(x)} and R2={(x,y)∣a⩽y⩽b,g(y)⩽x⩽h(y)}, where g(x), h(x), g(y) and h(y) are linear functions, is given from (x,y) space to a square in (ξ,η) space, S: {(ξ,η)∣0⩽ξ⩽1,0⩽η⩽1}. Generlized Gaussian quadrature nodes and weights introduced by Ma et.al. in 1997 are used in the product formula presented in this paper to evaluate the integral over S, as it is proved to give more accurate results than the classical Gauss Legendre nodes and weights. The method can be used to integrate a wide class of functions including smooth functions and functions with end-point singularities, over any two-dimensional region, bounded by linear sides. The performance of the method is illustrated for different functions over different two-dimensional regions with numerical examples.