  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 $u$.
