Optimal Order Convergence Implies Numerical Smoothness

TL;DR

This paper establishes that numerical smoothness, measured through derivative jumps and differences, is a necessary condition for achieving optimal convergence in piecewise polynomial approximations of PDEs and ODEs.

Contribution

It provides rigorous definitions of numerical smoothness and proves its necessity for optimal order convergence on various mesh types in multiple dimensions.

Findings

Numerical smoothness is necessary for optimal convergence.

Defined measures of smoothness include scaled derivative jumps and interior differences.

Validated on quasi-uniform meshes in 2D and 3D.

Abstract

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.