Symmetric Gauss Legendre quadrature formulas for composite numerical integration over a triangular surface

Abstract

This paper first presents a Gauss Legendre quadrature method for numerical integration of I ¼ R R T f ðx; yÞ dx dy, where f(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 two 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 integral R R S f ðxðn; gÞ; yðn; gÞÞ oðx; yÞ oðn;gÞ dn dg, where S is now the 2-square in (n, g) space: {(n, g)j À 1 6 n, g 6 1}. We then apply the one dimensional Gauss Legendre quadrature rules in n and g variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. We then propose the discretisation of the standard triangular surface T into n 2 right isosceles triangular surfaces T i (i = 1(1)n 2 ) each of which has an area equal to 1/(2n 2 ) units. We have again shown that the use of affine transformation over each T i and the use of linearity property of integrals lead to the result:where H ðX ; Y Þ ¼ P nÂn i¼1 f ðx i ðX ; Y Þ; y i ðX ; Y ÞÞ and x = x i (X, Y) and y = y i (X, Y) refer to affine transformations which map each T i in (x, y) space into a standard triangular surface T in (X, Y) space. We can now apply Gauss Legendre quadrature formulas which are derived earlier for I to evaluate the integral I ¼ 1 n 2 RR T H ðX ; Y Þ dX dY . We observe that the above procedure which clearly amounts to Composite Numerical Integration over T and it converges to the exact value of the integral RR T f ðx; yÞ dx dy, for sufficiently large value of n, even for the lower order Gauss Legendre quadrature rules. We have demonstrated this aspect by applying the above explained Composite Numerical Integration method to some typical integrals.

Optimal Subparametric Finite Elements for Elliptic Partial Differential Equations Using Higher-Order Curved Triangular Elements

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A separation between the boundary shape and the boundary functions in the parametric integral equation system for the 3D Stokes equation

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New fully symmetric and rotationally symmetric cubature rules on the triangle using minimal orthonormal bases

“…This is perhaps the most studied cubature domain, with a correspondingly large body of literature a selection of which is presented here. While rules of degree up to 20, thus covering most cases of practical interest, were progressively developed by 1985 [1,4,5,6], this is still an active field [7,8,9,10,11,12,13,14,15,16]. This happens for two distinct reasons, the first being that different applications require different properties of the cubature rules; the previously cited work for example focuses only on fully symmetric rules (which are also the easier to determine), while only a few works consider rotationally symmetric [17,12] or asymmetric [18,19] rules.…”