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

摘要

This paper first presents a Gauss Legendre quadrature method for numerical integration of I=∫∫Tf(x,y)dxdy, 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 Cartesian two dimensional (x, y) space. We then use a transformation x = x(ξ, η), y = y(ξ, η) to change the integral I to an equivalent integral ∫∫Sf(x(ξ,η),y(ξ,η))∂(x,y)∂(ξ,η)dξdη, where S is now 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 rule with new weight coefficients and new sampling points. We then 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/(2n2) 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:I=∑i=1n×n∫∫Tif(x,y)dxdy=1n2∫∫TH(X,Y)dXdY,where H(X,Y)=∑i=1n×nf(xi(X,Y),yi(X,Y)) and x = xi(X, Y) and y = yi(X, Y) refer to affine transformations which map each Ti 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=1n2∫∫TH(X,Y)dXdY. 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 ∫∫Tf(x,y)dxdy, 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.