An estimate of accuracy for interpolant numerical solutions of a PDE problem

TL;DR

This paper provides an accuracy estimate for polynomial interpolant solutions of nonlinear PDEs, demonstrating they can be accurate and computationally efficient compared to fine numerical solutions under certain conditions.

Contribution

It introduces a theoretical error estimate for interpolant solutions of PDEs, highlighting their potential for efficient approximation with reduced computational cost.

Findings

Interpolant solutions can closely approximate fine numerical solutions under certain hypotheses.

The computational cost of interpolant solutions is significantly lower than that of fine solutions.

Applications to linear and periodic PDE cases illustrate the method's effectiveness.

Abstract

In this paper we present an estimate of accuracy for a piecewise polynomial approximation of a classical numerical solution to a non linear differential problem. We suppose the numerical solution U is computed using a grid with a small linear step and interval time Tu, while the polynomial approximation V is an interpolation of the values of a numerical solution on a less fine grid and interval time Tv << Tu. The estimate shows that the interpolant solution V can be, under suitable hypotheses, a good approximation and in general its computational cost is much lower of the cost of the fine numerical solution. We present two possible applications to linear case and periodic case.