Gaussian Quadrature Rule for Triangle and Tetrahedron

Gaussian Quadrature rule for Triangle and Tetrahedron Qikun Wu, Liuxing Shen Introduction We demonstrate the position of the Gaussian points in 2D and 3D case, and finished task 2.

1. two dimension case in two dimension case, we will talk about square and triangle. 1.1 tensor products of the one dimensional formula intuitively, if we directly use tensor product of one-dimensional case, we can have a ‘Gaussian Quadrature’ scheme in two dimension, which is illustrated as follows:

 

1

1

1 1 1

f ( , )d d  

1

1



1

1

f ( , )d  d



n 1  n      H i f (i , ) d   H i  f (i , )d 1 1  i 1  i 1 n  n  n n   H i   H j f (i , j )    H i H j f (i , j ) i 1  j 1  i 1 j 1

And this tensor product can extend to arbitrary dimension:

 

n

1

1

1 1 n



1

1

f ( x1 , x2

n i1

xm )d x1dx2

dxm xm( im ) )

 

i1 1 i2 1

im 1

H

Hi2

Him f ( x1( i1 ) , x2( i2 )

However, we can see that this quadrature is not based on some ‘orthogonal polynomial in 2D’, which is not an optimal solution, so we can expect that we can choose less point to reach the same accuracy. So we should let the quadrature scheme with the following form:

I 

1

1 1



1

f ( , )d d  Wi f (i ,i )

i 1

n

1.2 non-tensor-product formula As stated before, the non-tensor product form can be made using only 7 points to achieve the same accuracy while the tensor form need 9 points. However, since the orthogonal polynomials are unknown in two and three dimension, these non-tensor-product form are complicated to derive. The corresponding coefficients are usually determined by method of underdetermined coefficients. We shall show the example of triangle in the next section. 1.3 quadrature rule in triangle We now discuss the scenario where we will do integral on a canonical triangle, where the coordinates of the vertexes are [0,0], [0,1], [1,0]. We first consider the simplest case:

 f ( , )d d  W f ( , )  E

1 1 1 ...