Application of Taylor's formula to polynomial approximation of a function of two variables with large gradients

The problem of approximating a function of two variables with large gradients by polynomials based on the Taylor formula is investigated. It is assumed that the decomposition of the function in the form of a sum of regular and boundary layer components is valid. The boundary layer component is known with an accuracy of up to a factor and is responsible for large gradients of the function. Such a decomposition is valid for the solution of singularly perturbed elliptic problem. The problem is that approximating such a function by polynomials based on the Taylor formula can lead to significant errors due to the presence of the boundary layer component. A formula for approximating a function is developed, using the Taylor formula and, by construction, being exact on the boundary layer component of the given function of two variables. It is proved that the error estimate of the constructed formula depends on the partial derivatives of the regular component and does not depend on the derivatives of the boundary layer component, which significantly increases the accuracy of approximating the function by polynomials.