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Abstract <p> We consider approximation of a function of two variables with large gradients ina neighborhood of a point on the basis of Taylor’s formula. If the derivatives of this function arenot bounded by a constant then the error of such an approximation can be significant. Werepresent the function as the sum of the regular and boundary layer components. Sucha 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 toa factor and causes large gradients of the function.</p> <p>We suggest to improve the accuracy of approximation on the basis of Taylor’sformula 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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年3月1日

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  • A. I. Zadorin

摘要

Abstract

We consider approximation of a function of two variables with large gradients ina neighborhood of a point on the basis of Taylor’s formula. If the derivatives of this function arenot bounded by a constant then the error of such an approximation can be significant. Werepresent the function as the sum of the regular and boundary layer components. Sucha 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 toa factor and causes large gradients of the function.

We suggest to improve the accuracy of approximation on the basis of Taylor’sformula 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.