Publication

Abstract : In this paper we first present a Gauss–Legendre quadrature rule for the evaluation of View the MathML source, where f(x, y, z) is an analytic function in x, y, z and T is the standard tetrahedral region: {(x, y, z)∣0 ⩽ x, y, z ⩽ 1, x + y + z ⩽ 1} in three space (x, y, z). We then use a transformation x = x(ξ, η, ζ), y = y(ξ, η, ζ) and z = z(ξ, η, ζ ) to change the integral into an equivalent integral View the MathML source over the standard 2-cube in (ξ, η, ζ) space: {(ξ, η, ζ)∣ −1 ⩽ ξ, η, ζ ⩽ 1}. We then apply the one-dimensional Gauss–Legendre quadrature rules in ξ, η and ζ variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. Then a second Gauss–Legendre quadrature rule of composite type is obtained. This rule is derived by discretising the tetrahedral region T into four new tetrahedra View the MathML source (i = 1, 2, 3, 4) of equal size which are obtained by joining the centroid of T, c = (1/4, 1/4, 1/4) to the four vertices of T . By use of the affine transformations defined over each View the MathML source and the linearity property of integrals leads to the result: