7 research outputs found
The enriched Crouzeix-Raviart elements are equivalent to the Raviart-Thomas elements
For both the Poisson model problem and the Stokes problem in any dimension,
this paper proves that the enriched Crouzeix-Raviart elements are actually
identical to the first order Raviart-Thomas elements in the sense that they
produce the same discrete stresses. This result improves the previous result in
literature which, for two dimensions, states that the piecewise constant
projection of the stress by the first order Raviart-Thomas element is equal to
that by the Crouzeix-Raviart element. For the eigenvalue problem of Laplace
operator, this paper proves that the error of the enriched Crouzeix-Raviart
element is equivalent to that of the Raviart-Thomas element up to higher order
terms
Explicit estimation of error constants appearing in non-conforming linear triangular finite element method
summary:The non-conforming linear () triangular FEM can be viewed as a kind of the discontinuous Galerkin method, and is attractive in both the theoretical and practical purposes. Since various error constants must be quantitatively evaluated for its accurate a priori and a posteriori error estimates, we derive their theoretical upper bounds and some computational results. In particular, the Babuška-Aziz maximum angle condition is required just as in the case of the conforming triangle. Some applications and numerical results are also included to see the validity and effectiveness of our analysis
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