On the application of two symmetric Gauss Legendre quadrature rules for composite numerical integration over a triangular surface

摘要

This paper first presents a Gauss Legendre quadrature rule for the evaluation of I = ∫ ∫T f (x, y) d x d y, where f (x, y) is an analytic function in x, y and T is the standard triangular surface: {(x, y) | 0 ≤ x, y ≤ 1, x + y ≤ 1} in the two space (x, y). We transform this integral into an equivalent integral ∫ ∫S f (x (ξ, η), y (ξ, η)) frac(∂ (x, y), ∂ (ξ, η)) d ξ d η where S is the 2-square in (ξ, η) space: {(ξ, η) | - 1 ≤ ξ, η ≤ 1}. We then apply the one-dimensional Gauss Legendre quadrature rules in ξ and η variables to arrive at an efficient Quadrature rules 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 T into three new triangles TiC (i = 1, 2, 3) of equal size which are obtained by joining centroid of T, C = (1 / 3, 1 / 3) to the three vertices of T. By use of affine transformations defined over each TiC and the linearity property of integrals leads to the result:I = underover(∑, i = 1, 3) ∫ ∫TiC f (x, y) d x d y = frac(1, 3) ∫ ∫T G (X, Y) d X d Y,where G (X, Y) = ∑i = 1n × n f (xiC (X, Y), yiC (X, Y)) and x = xiC (X, Y) and y = yiC (X, Y) refer to affine transformations which map each TiC into T the standard triangular surface. We then write ∫ ∫T G (X, Y) d X d Y = ∫ ∫S G (X (ξ, η), Y (ξ, η)) frac(∂ (X, Y), ∂ (ξ, η)) d ξ d η and a composite rule of integration is thus obtained. We next propose the discretisation of the standard triangular surface T into n2 right isosceles triangular surfaces Ti(i = 1 (1) n2) each of which has an area equal to 1 / (2 n2) units. We have again shown that the use of affine transformation over each Ti and the use of linearity property of integrals lead to the result:∫ ∫T f (x, y) d x d y = underover(∑, i = 1, n × n) ∫ ∫Ti f (x, y) d x d y = frac(1, n2) ∫ ∫T H (X, Y) d X d Y,where H (X, Y) = ∑i = 1n × n f (xi (X, Y), yi (X, Y)) and x = xi (X, Y), y = yi (X, Y) refer to affine transformations which map each Ti in (x, y) space into T a standard triangular surface T in the (x, y) space. We can now apply the two rules earlier derived to the integral ∫ ∫T H (X, Y) d X d Y, this amounts to application of composite numerical integration of T into n2 and 3n2 triangles of equal sizes respectively. We can now apply the rules, which are derived earlier to the evaluation of the integral, ∫ ∫T f (x, y) d x d y and each of these procedures converges to the exact value of the integral ∫ ∫T f (x, y) d x d y for sufficiently large value of n and the convergence is much faster for higher order rules. We have demonstrated this aspect by applying the above composite integration method to some typical integrals. © 2007 Elsevier Inc. All rights reserved.