  It is a widely held view that analytical integration is more accurate than
the numerical one. In some special cases, however, numerical integration can be
more advantageous than analytical integration. In our paper we show this
benefit for the case of electric potential and field computation of charged
triangles and rectangles applied in the boundary element method (BEM).
Analytical potential and field formulas are rather complicated (even in the
simplest case of constant charge densities), they have usually large
computation times, and at field points far from the elements they suffer from
large rounding errors. On the other hand, Gaussian cubature, which is an
efficient numerical integration method, yields simple and fast potential and
field formulas that are very accurate far from the elements. The simplicity of
the method is demonstrated by the physical picture: the triangles and
rectangles with their continuous charge distributions are replaced by discrete
point charges, whose simple potential and field formulas explain the higher
accuracy and speed of this method. We implemented the Gaussian cubature method
for the purpose of BEM computations both with CPU and GPU, and we compare its
performance with two different analytical integration methods. The ten
different Gaussian cubature formulas presented in our paper can be used for
arbitrary high-precision and fast integrations over triangles and rectangles.
