Cubature of integrands containing derivatives

Summary. We present a new technique for the numerical integration over R, a square or triangle, of an integrand of the form (ru) T B(rv). This uses only function values of u, B, and v, avoiding explicit di erentiation, but is suitable only when the integrand function is regular over R. The technique is analogous to Romberg integration, since it is based on using a sequence of very simple discretizations 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 speci c cost-e ective 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