Generalized Composite Numerical Integration Rule Over a Polygon Using Gaussian Quadrature

Abstract

The aim of this paper is to evaluate double integrals over a polygon exploiting coordinate transformation. At first any polygon with m-sides is decomposed into triangles. Then each triangle is transformed into a standard triangular finite element using the basis functions in local space. Then the standard triangle is decomposed into right isosceles triangles with side lengths , and thus composite numerical integration is employed. In addition, the affine transformation over each decomposed triangle and the use of linearity property of integrals are applied. Finally, each isosceles triangle is transformed into a 2-sqare finite element to compute new sampling points and corresponding weight coefficients, using2 s s point's conventional Gaussian quadrature, which are applied again to evaluate the double integral. We demonstrate some numerical examples through the proposed method.

Testing wave turbulence theory for Gross-Pitaevskii system

“…One can see that expression (31) decays as t → ∞ for any fixed p, provided that θ → ∞. This condition for approaching to Gaussianity is equivalent to the respective condition for the PDF solution, (27).…”

“…At present, we do not have an explanation for this behaviour. the reference square [−1, 1] 2 (see [27] for the additional details), and tensor products of the modified Clenshaw-Curtis formulas are used to compute the integrals. It should be noted that, in contrast to the formulation considered in [6], here we have a bounded domain of integration, which somewhat simplifies the problem.…”