A local algorithm for smooth approximation of approximate difference and nonsmooth variational solutions

A local algorithm for smooth approximation is proposed for approximate solutions of the one-dimensional problems which are obtained by difference methods or variational algorithms on the basis of piecewise smooth test functions. The algorithm is aimed at approximate solutions which are obtained with an error $O(h^\nu)$, $\nu=1,2$. The algorithm has been primordially parallelized and is utmost easy both in theoretical and in practical aspects. A result of the local approximation is a twice continuous differentiable function which keeps geometric properties of the initial approximate solution. Certain advantages of the proposed algorithm as compared to the cubic splines are shown.