Raising the order of multivariate approximation schemes using supplementary derivative data

We propose a generic procedure to raise the approximation order of multivariate approximation schemes using supplementary derivative data. The procedure applies to all schemes that reproduce polynomials to a certain degree, including most common types of (quasi-) interpolation and moving least-squares. For an approximation scheme of order m and a dataset that provides n supplementary orders of derivative data, the procedure results in an approximation order of m+n. This is achieved using a modification of the Taylor expansion, the reduced dual Taylor expansion, that is applied to the data prior to the evaluation of the scheme. The procedure is easy to implement in existing schemes and is expected to be useful immediately in a wide range of applications.