### A reduced basis element method for the steady
Stokes problem

**Auteur(s): **

**
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**Alf Emil Løvgren, Yvon Maday and Einar M. Rønquist**

**Le document est une prépublication**
**Code(s) de Classification MSC:**

**Abstract: ** The reduced basis element method is a new approach for approxi-
mating the solution of problems described by partial differential equa-
tions. The method takes its roots in domain decomposition methods
and reduced basis discretizations. The basic idea is to first decom-
pose the computational domain into a series of subdomains that are
deformations of a few reference domains (or generic computational
parts). Associated with each reference domain are precomputed solu-
tions corresponding to the same governing partial differential equation,
but solved for different choices of deformations of the reference sub-
domains and mapped onto the reference shape. The approximation
corresponding to a new shape is then taken to be a linear combina-
tion of the precomputed solutions, mapped from the reference domain
for the part to the actual domain. We extend earlier work in this
direction to solve incompressible fluid flow problems governed by the
steady Stokes equations. Particular focus is given to satisfying the
inf-sup condition, to a posteriori error estimation, and to "gluing" the
local solutions together in the multidomain case.

**Mots Clés:** *;*

**Date:** 2004-12-02