1,801 research outputs found
Analysis of Compatible Discrete Operator Schemes for the Stokes Equations on Polyhedral Meshes
Compatible Discrete Operator schemes preserve basic properties of the
continuous model at the discrete level. They combine discrete differential
operators that discretize exactly topological laws and discrete Hodge operators
that approximate constitutive relations. We devise and analyze two families of
such schemes for the Stokes equations in curl formulation, with the pressure
degrees of freedom located at either mesh vertices or cells. The schemes ensure
local mass and momentum conservation. We prove discrete stability by
establishing novel discrete Poincar\'e inequalities. Using commutators related
to the consistency error, we derive error estimates with first-order
convergence rates for smooth solutions. We analyze two strategies for
discretizing the external load, so as to deliver tight error estimates when the
external load has a large irrotational or divergence-free part. Finally,
numerical results are presented on three-dimensional polyhedral meshes
A posteriori error estimation for the Stokes problem: Anisotropic and isotropic discretizations
The paper presents a posteriori error estimators for the stationary Stokes problem. We consider anisotropic finite element discretizations (i.e. elements with very large aspect ratio) where conventional, isotropic error estimators fail.
Our analysis covers two- and three-dimensional domains, conforming and nonconforming discretizations as well as different elements.
This large variety of settings requires different approaches and results in different estimators. Furthermore many examples of finite element pairs that are covered by the analysis are presented.
Lower and upper error bounds form the main result with minimal assumptions on the elements. The lower error bound is uniform with respect to the mesh anisotropy with the exception of nonconforming 3D discretizations made of pentahedra or hexahedra. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation.
In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. Some of the corresponding results seem to be novel (in particular for 3D domains), and cover element pairs of practical importance.
The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimators
Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications
We develop a constructive piecewise polynomial approximation theory in
weighted Sobolev spaces with Muckenhoupt weights for any polynomial degree. The
main ingredients to derive optimal error estimates for an averaged Taylor
polynomial are a suitable weighted Poincare inequality, a cancellation property
and a simple induction argument. We also construct a quasi-interpolation
operator, built on local averages over stars, which is well defined for
functions in . We derive optimal error estimates for any polynomial degree
on simplicial shape regular meshes. On rectangular meshes, these estimates are
valid under the condition that neighboring elements have comparable size, which
yields optimal anisotropic error estimates over -rectangular domains. The
interpolation theory extends to cases when the error and function regularity
require different weights. We conclude with three applications: nonuniform
elliptic boundary value problems, elliptic problems with singular sources, and
fractional powers of elliptic operators
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