Numerical Integration over Curved Domains Using Convex Quadrangulations And Gauss Legendre Quadrature Rules

Abstract

This paper presents a numerical integration formula for the evaluation of where and is any curved domain in . That is a closed domain with boundary composed of N oriented piecewise curved segments with end points , and . We Join each of these curved segments to a reference point interior to the domain . This creates N triangles ) in and each of these triangles have one curved side and two straight sides. We transform each into a standard triangle T which also transforms the integrand to = . We then divide T into right isosceles triangles of side lengths 1/m units. These triangles will be finally divided into three special quadrilaterals . This process can be expressed as where represent the transformed forms of the integrand over the domains T, and . We approximate the curved segments by a parabolic arc which passes through the four points of the curved segment, the two end points , and the two intermediate points of . Proposed numerical integration formula is applied to integrate over a curved domain in the shape of lunar model for complicated integrands..