## Supercloseness of orthogonal projections onto nearby finite element spaces

^{1}
Computational and Mathematical Engineering, Stanford
University, Stanford,
CA,
USA

egawlik@stanford.edu

^{2}
Mechanical Engineering, Stanford University,
Stanford, CA, USA

lewa@stanford.edu

Received:
11
March
2014

Revised:
2
July
2014

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*^{γ}),
where *γ* 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*.

Mathematics Subject Classification: 65N30 / 65N15

Key words: Superconvergence / orthogonal projection / elliptic projection / *L*^{2}-projection

*© EDP Sciences, SMAI, 2015*