148 research outputs found
Comparison results for the Stokes equations
This paper enfolds a medius analysis for the Stokes equations and compares
different finite element methods (FEMs). A first result is a best approximation
result for a P1 non-conforming FEM. The main comparison result is that the
error of the P2-P0-FEM is a lower bound to the error of the Bernardi-Raugel (or
reduced P2-P0) FEM, which is a lower bound to the error of the P1
non-conforming FEM, and this is a lower bound to the error of the MINI-FEM. The
paper discusses the converse direction, as well as other methods such as the
discontinuous Galerkin and pseudostress FEMs.
Furthermore this paper provides counterexamples for equivalent convergence
when different pressure approximations are considered. The mathematical
arguments are various conforming companions as well as the discrete inf-sup
condition
Discontinuous Galerkin Methods for the Biharmonic Problem on Polygonal and Polyhedral Meshes
We introduce an -version symmetric interior penalty discontinuous
Galerkin finite element method (DGFEM) for the numerical approximation of the
biharmonic equation on general computational meshes consisting of
polygonal/polyhedral (polytopic) elements. In particular, the stability and
-version a-priori error bound are derived based on the specific choice of
the interior penalty parameters which allows for edges/faces degeneration.
Furthermore, by deriving a new inverse inequality for a special class {of}
polynomial functions (harmonic polynomials), the proposed DGFEM is proven to be
stable to incorporate very general polygonal/polyhedral elements with an
\emph{arbitrary} number of faces for polynomial basis with degree . The
key feature of the proposed method is that it employs elemental polynomial
bases of total degree , defined in the physical coordinate
system, without requiring the mapping from a given reference or canonical
frame. A series of numerical experiments are presented to demonstrate the
performance of the proposed DGFEM on general polygonal/polyhedral meshes
Adaptive Nonconforming Finite Element Approximation of Eigenvalue Clusters
This paper analyses an adaptive nonconforming finite element method for eigenvalue clusters of self-adjoint operators and proves optimal convergence rates (with respect to the concept of nonlinear approximation classes) for the approximation of the invariant subspace spanned by the eigenfunctions of the eigenvalue cluster. Applications include eigenvalues of the Laplacian and of the Stokes system
Instance optimal Crouzeix-Raviart adaptive finite element methods for the Poisson and Stokes problems
We extend the ideas of Diening, Kreuzer, and Stevenson [Instance optimality
of the adaptive maximum strategy, Found. Comput. Math. (2015)], from conforming
approximations of the Poisson problem to nonconforming Crouzeix-Raviart
approximations of the Poisson and the Stokes problem in 2D. As a consequence,
we obtain instance optimality of an AFEM with a modified maximum marking
strategy
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