  It is natural to expect the following loosely stated approximation principle
to hold: a numerical approximation solution should be in some sense as smooth
as its target exact solution in order to have optimal convergence. For
piecewise polynomials, that means we have to at least maintain numerical
smoothness in the interiors as well as across the interfaces of cells or
elements. In this paper we give clear definitions of numerical smoothness that
address the across-interface smoothness in terms of scaled jumps in derivatives
[9] and the interior numerical smoothness in terms of differences in derivative
values. Furthermore, we prove rigorously that the principle can be simply
stated as numerical smoothness is necessary for optimal order convergence. It
is valid on quasi-uniform meshes by triangles and quadrilaterals in two
dimensions and by tetrahedrons and hexahedrons in three dimensions. With this
validation we can justify, among other things, incorporation of this principle
in creating adaptive numerical approximation for the solution of PDEs or ODEs,
especially in designing proper smoothness indicators or detecting potential
non-convergence and instability.
