Abstract
Interpolation of functions on the basis of Lagrange’s polynomials is widely used. However in the case when the function has areas of large gradients, application of polynomials of Lagrange leads to essential errors. It is supposed that the function of one variable has the representation as a sum of regular and boundary layer components. It is supposed that derivatives of a regular component are bounded to a certain order, and the boundary layer component is a function, known within a multiplier; its derivatives are not uniformly bounded. A solution of a singularly perturbed boundary value problem has such a representation. Interpolation formulas, which are exact on a boundary layer component, are constructed. Interpolation error estimates, uniform in a boundary layer component and its derivatives are obtained. Application of the constructed interpolation formulas to creation of formulas of the numerical differentiation and integration of such functions is investigated.
Highlights
Lagrange polynomials for interpolation of functions are widely used
Interpolation formulas, which are exact on a boundary layer component, are constructed
Application of the constructed interpolation formulas to creation of formulas of the numerical differentiation and integration of such functions is investigated
Summary
Lagrange polynomials for interpolation of functions are widely used. According to [1], application of Lagrange polynomials for interpolation of functions with large gradients can lead to large interpolation errors. We suppose that the function under interpolation has the representation as the sum of regular and boundary layer components. Derivatives of the regular component are bounded up to some order. The boundary layer component is known to within a multiplier and has large gradients. It is known that the solution of a singularly perturbed boundary value problem has such representation [2]. We construct the interpolation formula which is exact on the boundary layer component. We use the constructed interpolation formula for the creation of formulas of numerical differentiation and integration of functions with the boundary layer component. Through the paper C and Cj denote generic positive constants independent of ε and mesh size
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