Cubature of Integrands Containing Derivatives

We present a new technique for the numerical integration over R a square or triangle of an integrand of the form ru B rv This uses only function values of u B and v avoiding ex plicit 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 discretiza tions J m m 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 pro vide 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