On the application of two symmetric Gauss legendre quadrature rules for composite numerical integration over a tetrahedral region

In this paper, we first present a Gauss Legendre Quadrature rule for the evaluation of I = â« T f(x,y,z) dxdydz, 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 the transformations x = x(ξ,η,ζ), y = y(ξ,η,ζ) and z = z(ξ,η,ζ) to change the integral I into an equivalent integral I = â« -1 1 â« -1 1 â« -1 1 f(x(ξ,η,ζ),y(ξ,η,ζ),z(ξ,η,ζ)) â(x,y,z)/â(ξ,η,ζ) dη dζ< 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 rule 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 T i<sup>c</sup> (i = 1,2,3,4) of equal size, which are obtained by joining centroid of T, c = (1/4, 1/4, 1/4) to the four vertices of T. Use of the affine transformations defined over each T i<sup>c</sup> and the linearity property of integrals leads to the result: </p><p>where refer to the affine transformations, which map each T i<sup>c</sup> into the standard tetrahedral region T. We then write and a composite rule of integration is thus obtained. We next propose the discretisation of the standard tetrahedral region T into p<sup>3</sup> tetrahedra T i (i = 1 (1) p<sup>3</sup>), each of which has volume equal to 1/(6p<sup>3</sup>) units. We have again shown that the use of affine transformations over each T i and the use of linearity property of integrals leads to the result: where refer to the affine transformations, which map each T i in (x<sup>(α,p)</sup>, y<sup>(α,p)</sup>, z<sup>(α,p)</sup>) space into a standard tetrahedron T in the (X,Y,Z) space. We can now apply the two rules earlier derived to the integral â« TH(X,Y,Z)dXdYdZ; this amounts to the application of composite numerical integration of T into p<sup>3</sup> and 4p<sup>3</sup> tetrahedra of equal volume. We have demonstrated this aspect by applying the above composite integration method to some typical triple integrals