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

We then writeI=∫∫T∫G(X,Y,Z)dXdYdZ=∫01∫01-ξ∫01-ξ-ηG(X(ξ,η,ζ),Y(ξ,η,ζ),Z(ξ,η,ζ))∂(X,Y,Z)∂(ξ,η,ζ)dξdηdζand a composite rule of integration is thus obtained. We next propose the discretisation of the standard tetrahedral region T into p3 tetrahedra Ti (i = 1(1)p3) each of which has volume equal to 1/(6p3) units. We have again shown that the use of affine transformations over each Ti and the use of linearity property of integrals leads to the result:∫∫T∫f(x,y,z)dxdydz=∑i=1p3∫∫Tic∫f(x,y,z)dxdydz=∑α=1p3∫∫Tα(p)∫f(x(α,p),y(α,p),z(α,p))dx(α,p)dy(α,p)dz(α,p)=1p3∫∫T∫H(X,Y,Z)dXdYdZ,whereH(X,Y,Z)=∑α=1P3f(x(α,P)(X,Y,Z),y(α,P)(X,Y,Z),z(α,P)(X,Y,Z)),x(α,p)=x(α,p)(X,Y,Z),y(α,p)=y(α,p)(X,Y,Z)andz(α,p)=z(α,p)(X,Y,Z)refer to the affine transformations which map each Ti in (x(α,p), y(α,p), z(α,p)) space into a standard tetrahedron T in the (X, Y, Z) space. We can now apply the two rules earlier derived to the integral ∫∫T∫H(X,Y,Z)dXdYdZ, this amounts to the application of composite numerical integration of T into p3 and 4p3 tetrahedra of equal sizes. We have demonstrated this aspect by applying the above composite integration method to some typical triple integrals.