A Lagrange interpolation with preprocessing to nearly eliminate oscillations

Abstract

This work is concerned with the interpolation of a function f when using a low number of interpolation points, as required by the finite element method for solving PDEs numerically. The function f is assumed to have a jump or a steep derivative, and our goal is to minimize the oscillations produced by the Gibbs phenomenon while preserving the approximation properties for smoother functions. This is achieved by interpolating the transform f^=g∘f using Lagrange polynomials, where g is a rational transformation chosen by minimizing a suitable functional depending on the values of f. The mapping g is monotonic and constructed to possess boundary layers that remove the Gibbs phenomenon. No previous knowledge of the location of the jump is required. The extension to functions of several variables is straightforward, of which we provide several examples. Finally, we show how the interpolation fits the finite element method and compare it with known strategies.