12 research outputs found
Superconvergence Using Pointwise Interpolation in Convection-Diffusion Problems
Considering a singularly perturbed convection-diffusion problem, we present
an analysis for a superconvergence result using pointwise interpolation of
Gau{\ss}-Lobatto type for higher-order streamline diffusion FEM.
We show a useful connection between two different types of interpolation,
namely a vertex-edge-cell interpolant and a pointwise interpolant. Moreover,
different postprocessing operators are analysed and applied to model problems.Comment: 19 page
Supercloseness of Orthogonal Projections onto Nearby Finite Element Spaces
We derive upper bounds on the difference between the orthogonal projections
of a smooth function 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 , where is a nonnegative scalar and
is the mesh spacing. The projector may be, for example, the orthogonal
projector with respect to the - or -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 . We present numerical examples to illustrate
these superconvergent estimates and verify the necessity of the regularity
assumptions on
Supercloseness and asymptotic analysis of the Crouzeix-Raviart and enriched Crouzeix-Raviart elements for the Stokes problem
For the Crouzeix-Raviart and enriched Crouzeix-Raviart elements, asymptotic
expansions of eigenvalues of the Stokes operator are derived by establishing
two pseudostress interpolations, which admit a full one-order supercloseness
with respect to the numerical velocity and the pressure, respectively. The
design of these interpolations overcomes the difficulty caused by the lack of
supercloseness of the canonical interpolations for the two nonconforming
elements, and leads to an intrinsic and concise asymptotic analysis of
numerical eigenvalues, which proves an optimal superconvergence of eigenvalues
by the extrapolation algorithm. Meanwhile, an optimal superconvergence of
postprocessed approximations for the Stokes equation is proved by use of this
supercloseness. Finally, numerical experiments are tested to verify the
theoretical results