Application of Taylor’s Formula to Polynomial Approximation ofa Function of Two Variables with Large Gradients

We consider approximation of a function of two variables with large gradients in a neighborhood of a point on the basis of Taylor’s formula. If the derivatives of this function are not bounded by a constant then the error of such an approximation can be significant. We represent the function as the sum of the regular and boundary layer components. Such a representation exists, for example, for solutions of singularly perturbed elliptic problems. The boundary layer component is regarded as a function of a general form. It is determined up to a factor and causes large gradients of the function. We suggest to improve the accuracy of approximation on the basis of Taylor’s formula by constructing formulas that are exact on the boundary layer component. We prove that, in this case, the error estimate is independent of the derivatives of the boundary layer component.