Necessity of numerical smoothness

Abstract: Numerical solutions of differential equations are usually not smooth functions. However, they should resemble the smoothness of the corresponding real solutions in one way or another. In two of our papers, a kind of spacial smoothness indicators was defined and subsequently applied on the {\it a posteriori} error analysis. Here we prove that the boundedness of those smoothness indicators is actually a necessary condition for a piecewise polynomial function to approximate a smooth function with optimal convergence rate. This should help in validating the error analysis in the related papers. Moreover, the result of this paper provides an efficient practical method to detect the loss of convergence rate due to the lack of numerical smoothness, hence it serves as a criterion for the qualities of many numerical schemes.