Supercloseness of Orthogonal Projections onto Nearby Finite Element Spaces

TL;DR

This paper establishes superconvergent bounds on the difference between orthogonal projections of a smooth function onto two nearby finite element spaces, showing faster decay rates under mesh refinement.

Contribution

It derives novel superconvergence bounds for projections onto finite element spaces that are close in a measure-theoretic sense, applicable to common finite element settings.

Findings

Bounds are superconvergent under mild regularity assumptions.

The difference between projections decays faster than individual projection errors.

Numerical examples confirm theoretical estimates and regularity requirements.

Abstract

We derive upper bounds on the difference between the orthogonal projections of a smooth function $u$ onto two finite element spaces that are nearby, in the sense that the support of every shape function belonging to one but not both of the spaces is contained in a common region whose measure tends to zero under mesh refinement. The bounds apply, in particular, to the setting in which the two finite element spaces consist of continuous functions that are elementwise polynomials over shape-regular, quasi-uniform meshes that coincide except on a region of measure $O(h^\gamma)$, where $\gamma$ is a nonnegative scalar and $h$ is the mesh spacing. The projector may be, for example, the orthogonal projector with respect to the $L^2$- or $H^1$-inner product. In these and other circumstances, the bounds are superconvergent under a few mild regularity assumptions. That is, under mesh refinement, the two projections differ in norm by an amount that decays to zero at a faster rate than the amounts by which each projection differs from $u$. We present numerical examples to illustrate these superconvergent estimates and verify the necessity of the regularity assumptions on $u$.