89 research outputs found
A Unified Approach for Uzawa Algorithms
International audienceWe present a unified approach in analyzing Uzawa iterative algorithms for saddle point problems. We study the classical Uzawa method, the augmented Lagrangian method, and two versions of inexact Uzawa algorithms. The target application is the Stokes system, but other saddle point systems, e.g., arising from mortar methods or Lagrange multipliers methods, can benefit from our study. We prove convergence of Uzawa algorithms and find optimal rates of convergence in an abstract setting on finite-or infinite-dimensional Hilbert spaces. The results can be used to design multilevel or adaptive algorithms for solving saddle point problems. The discrete spaces do not have to satisfy the LBB stability condition
Notes on Finite Element Discretization for a Model Convection-Diffusion Problem
We present recent finite element numerical results on a model
convection-diffusion problem in the singular perturbed case when the convection
term dominates the problem. We compare the standard Galerkin discretization
using the linear element with a saddle point least square discretization that
uses quadratic test functions, trying to control and explain the non-physical
oscillations of the discrete solutions. We also relate the up-winding
Petrov-Galerkin method and the stream-line diffusion discretization method, by
emphasizing the resulting linear systems and by comparing appropriate error
norms. Some results can be extended to the multidimensional case in order to
come up with efficient approximations for more general singular perturbed
problems, including convection dominated models.Comment: 24 pages, 12 figure
Weighted Sobolev spaces and regularity for polyhedral domains
We prove a regularity result for the Poisson problem , u
|\_{\pa \PP} = g on a polyhedral domain \PP \subset \RR^3 using the \BK\
spaces \Kond{m}{a}(\PP). These are weighted Sobolev spaces in which the
weight is given by the distance to the set of edges \cite{Babu70,
Kondratiev67}. In particular, we show that there is no loss of
\Kond{m}{a}--regularity for solutions of strongly elliptic systems with
smooth coefficients. We also establish a "trace theorem" for the restriction to
the boundary of the functions in \Kond{m}{a}(\PP)
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