# Numerical integration on a triangle

1. May 22, 2008

### daudaudaudau

Hi.

To solve an integral equation I have to numerically integrate the scalar Green's function on a triangle having vertices $$(x,y) = (0,0), (1,0), (1,1)$$:

$$I = \int_0^1\int_0^x \frac{e^{-i 2\pi R}}{4\pi R} dy dx$$

where

$$R=\sqrt{(x-x')^2+(y-y')^2}$$

is the distance between a fixed point outside the triangle and the integration point inside the triangle. Does anyone have any experience in this area?

To evaluate this integral one can use precomputed weights and abcissae for triangular domains, and approximate the integral by a product rule, i.e.

$$I=\sum_{n=1}^N w_i f(x_i,y_i)$$

where $$f(x,y)$$ is the Green's function. But this will only be accurate when $$R$$ is sufficiently large. When $$R$$ is small, the integrand grows increasingly singular.

So I could use the product rule only when $$R$$ is larger than some constant, but I wonder if this is the smartest way of doing it. I guess this is a very typical problem encountered when solving any kind of real-world integral equation, but I have not been able to find any info about this problem.