Some High Degree Gauss Legendre Quadrature Formulas for Triangles

This paper presents a numerical integration formula for the evaluation of ( )

∬ ( )

where ( ) and is any polygonal domain in . That is a closed domain with

boundary composed of N oriented piecewise line segments ( ) with end points

( ), ( ) and ( ) (( ). We join each of these line segments to a reference point

( ) interior to the domain . This creates a coarse mesh of N triangles ( ) in and

each of these arbitrary triangles have three straight sides. These arbitrary triangles can be divided into

arbitrary triangles ( ,2,3,…. ). by using the triangular mesh generation scheme

developed in this paper. We transform each = (say) into a standard 1-square and then into a 2-square

which can be integrated by using Gauss Legendre quadraure rules.We obtain enhanced accuracy by

division of these arbitrary triangles into four arbitrary triangles (e=1,2,3,4) without refining the already

generated triangular mesh.We first derive three different integration formulas for the integral

∬ ( )

where (( ) ) Then we

establish the necessary numerical integration formulas which use the well known Gauss Legendre

quadrature rules. Proposed numerical integration formula is applied to integrals over triangular and

polygonal domains with complicated integrands.