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

In this paper we present an estimate of accuracy for a piecewise polynomial approximation of a classical numerical solution to a non linear di erential 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 ne 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 ne numerical solution. We present two possible applications to linear case and periodic case. 1 The problem Let [a; b], a < b, an interval on the real line R, and let C1([0;+1) [a; b]) the space of continuously di erentiable real functions de ned on [0;+1) [a; b]. Let F : C1([0;+1) [a; b])! R a continuously di erentiable real functional. Then, if u 2 C1([0;+1) [a; b]), the partial di erential equation, usually associated to hyperbolic conservation laws,