Symmetric Gauss Legendre quadrature formulae for composite numerical integration over a tetrahedral region.

摘要

This paper first presents a Gauss Legendre quadrature method for numerical integration of I ¼ R R\udT f ðx; yÞdxdy, where\udf(x,y) is an analytic function in x, y and T is the standard triangular surface: {(x,y)j0 6 x, y 6 1, x + y 6 1} in the Cartesian\udtwo dimensional (x,y) space. We then use a transformation x = x(n,g), y = y(n,g) to change the integral I to an equivalent\udintegral R R\udS f ðxðn; gÞ; yðn; gÞÞ oðx; yÞ\udoðn;gÞ dndg, where S is now the 2-square in (n,g) space: {(n,g)j 1 6 n,g 6 1}. We then\udapply the one dimensional Gauss Legendre quadrature rules in n and g variables to arrive at an efficient quadrature rule\udwith new weight coefficients and new sampling points. We then propose the discretisation of the standard triangular surface\udT into n2 right isosceles triangular surfaces Ti (i = 1(1)n2\ud) each of which has an area equal to 1/(2n2\ud) units. We have\udagain shown that the use of affine transformation over each Ti and the use of linearity property of integrals lead to the\udresult:\udI ¼ Xnn\udi¼1\udZ Z\udT i\udf ðx; yÞdxdy ¼ 1\udn2\udZ Z\udT\udHðX; Y ÞdX dY ;\udwhere HðX; Y Þ ¼ Pnn\udi¼1 f ðxiðX; Y Þ; yiðX; Y ÞÞ and x = xi(X,Y) and y = yi(X,Y) refer to affine transformations which map\udeach Ti in (x,y) space into a standard triangular surface T in (X,Y) space. We can now apply Gauss Legendre quadrature\udformulas which are derived earlier for I to evaluate the integral I ¼ 1\udn2\udRR\udT HðX; Y ÞdX dY . We observe that the above procedure\udwhich clearly amounts to Composite Numerical Integration over T and it converges to the exact value of the integral\udRR\udT f ðx; yÞdxdy, for sufficiently large value of n, even for the lower order Gauss Legendre quadrature rules. We have\uddemonstrated this aspect by applying the above explained Composite Numerical Integration method to some typical\udintegrals.