28 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
Quasi-optimal nonconforming methods for symmetric elliptic problems. I -- Abstract theory
We consider nonconforming methods for symmetric elliptic problems and
characterize their quasi-optimality in terms of suitable notions of stability
and consistency. The quasi-optimality constant is determined and the possible
impact of nonconformity on its size is quantified by means of two alternative
consistency measures. Identifying the structure of quasi-optimal methods, we
show that their construction reduces to the choice of suitable linear operators
mapping discrete functions to conforming ones. Such smoothing operators are
devised in the forthcoming parts of this work for various finite element
spaces