Cubature of integrands containing derivatives

Abstract.

We present a new technique for the numerical integration over $\cal R$, a square or triangle, of an integrand of the form $(\nabla u)^{\rm T} B (\nabla v)$. This uses only function values of $u$, $B$, and $v$, avoiding explicit differentiation, but is suitable only when the integrand function is regular over $\cal R$. The technique is analogous to Romberg integration, since it is based on using a sequence of very simple discretizations $\mbox{\rm J}^{(m)}, m = 1,2,3,...,$ of the required integral and applying extrapolation in $m$ to provide closer approximations. A general approach to the problem of constructing discretizations is given. We provide specific cost-effective discretizations satisfying familiar, but somewhat arbitrary guidelines. As in Romberg integration, when each component function in the integrand is a polynomial, this technique leads to an exact result.