Éditeur(s) : HAL CCSD Wiley
Résumé : International audience
In this paper, the generalized Stokes problem αu−νΔu+∇p=f, ∇u=0, is considered, where ν denotes the kinematic viscosity, α is a positive parameter proportional to the inverse of the time-step, and the source term f gathers the external forcing, the nonlinear term, and the explicit part of the linear term. The system is derived from a time discretization for the Navier-Stokes equation governing the motion of a two-dimensional incompressible fluid. A domain decomposition numerical method is developed for solving the system. The original domain is divided into several nonoverlapping subdomains, and in each subdomain the concerned equations are discretized by certain finite element methods. A new feature of the method is that the pressure p is required to be piecewise linear rather than piecewise constant, so that the nonphysical oscillations in the solution using piecewise linear pressure are eliminated. However, a difficulty in using piecewise linear pressure is that the Stokes problem in each subdomain involves Dirichlet boundary conditions on pressure at the interface, which is not a real Stokes problem. To avoid the difficulty, a modification is proposed which removes the continuity conditions on the pressure across the interface. In this way, the Stokes problem in each subdomain has only Dirichlet boundary conditions on the velocity. Formulations and discussions for both the methods are presented, which shows that the modified method is simpler than the original one. Numerical experiments are displayed to show the efficiency of the methods, especially that the modified method has the same accuracy as the original one.
ISSN: 0749-159X

hal-00730200
https://hal.archives-ouvertes.fr/hal-00730200

Éditeur(s) : HAL CCSD
Résumé : To appear in TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS (16 pages).
In the last two decades, many algebras of generalized functions have been constructed, particularly the so-called generalized Sobolev algebras. Our goal is to study the latter and some of their main properties. In this framework, we pose and solve a nonlinear degenerated Dirichlet problem with irregular data such as Dirac generalized functions.
https://hal.archives-ouvertes.fr/hal-00004903

Droits : info:eu-repo/semantics/OpenAccess
hal-00004903
https://hal.archives-ouvertes.fr/hal-00004903
https://hal.archives-ouvertes.fr/hal-00004903/document
https://hal.archives-ouvertes.fr/hal-00004903/file/sb_psn_pjm.pdf
ARXIV : math.GM/0505228

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
Starting from a locally convex metrisable topological space and from any asymptotic scale, we construct a generalized extension of this space. To those extensions, we associate Hausdorff topologies. We introduce the notion of a temperate map, with respect to a given asymptotic scale, between two locally convex metrisable semi-normed spaces. We show that such mappings extend in a canonical way to mappings between the respective generalized extensions. We give an application to nonlinear Dirichlet boundary value problems with singular data in the framework of generalized extensions
ISSN: 0026-9255

hal-01529549
https://hal.univ-antilles.fr/hal-01529549

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
In a previous work, one of the authors has studied a numerical treatment (by fully implicit discretizations) of a two-dimensional Navier-Stokes-like problem and has proved existence and convergence results for the resulting discretized systems with homogeneous Dirichlet boundary conditions. In this work, we propose some new preconditioned multilevel versions of inexact-Newton algorithms to solve these equations. We also develop another multilevel preconditioner for a nonlinear GMRES algorithm. All of the preconditioners are based on incremental unknowns formulations.
ISSN: 0096-3003

hal-00770269
https://hal.univ-antilles.fr/hal-00770269
DOI : 10.1016/S0096-3003(96)00304-9