A reduced basis element method for the steady Stokes problem

Alf Emil Løvgren; Yvon Maday; Einar M. Rønquist

ESAIM: Mathematical Modelling and Numerical Analysis (2006)

  • Volume: 40, Issue: 3, page 529-552
  • ISSN: 0764-583X

Abstract

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The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations. The basic idea is to first decompose 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 solutions corresponding to the same governing partial differential equation, but solved for different choices of deformations of the reference subdomains and mapped onto the reference shape. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the generic computational part to the actual computational part. 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.

How to cite

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Løvgren, Alf Emil, Maday, Yvon, and Rønquist, Einar M.. "A reduced basis element method for the steady Stokes problem." ESAIM: Mathematical Modelling and Numerical Analysis 40.3 (2006): 529-552. <http://eudml.org/doc/249699>.

@article{Løvgren2006,
abstract = { The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations. The basic idea is to first decompose 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 solutions corresponding to the same governing partial differential equation, but solved for different choices of deformations of the reference subdomains and mapped onto the reference shape. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the generic computational part to the actual computational part. 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. },
author = {Løvgren, Alf Emil, Maday, Yvon, Rønquist, Einar M.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Stokes flow; reduced basis; reduced order model; domain decomposition; mortar method; output bounds; a posteriori error estimators.; a posteriori error estimator; inf-sup condition},
language = {eng},
month = {7},
number = {3},
pages = {529-552},
publisher = {EDP Sciences},
title = {A reduced basis element method for the steady Stokes problem},
url = {http://eudml.org/doc/249699},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Løvgren, Alf Emil
AU - Maday, Yvon
AU - Rønquist, Einar M.
TI - A reduced basis element method for the steady Stokes problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2006/7//
PB - EDP Sciences
VL - 40
IS - 3
SP - 529
EP - 552
AB - The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations. The basic idea is to first decompose 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 solutions corresponding to the same governing partial differential equation, but solved for different choices of deformations of the reference subdomains and mapped onto the reference shape. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the generic computational part to the actual computational part. 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.
LA - eng
KW - Stokes flow; reduced basis; reduced order model; domain decomposition; mortar method; output bounds; a posteriori error estimators.; a posteriori error estimator; inf-sup condition
UR - http://eudml.org/doc/249699
ER -

References

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