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

Supercloseness of Orthogonal Projections onto Nearby Finite Element Spaces