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.

Keywords: Stokes flow, reduced basis, reduced order model, domain decomposition, mortar method, output bounds, a posteriori error estimators

@article{M2AN_2006__40_3_529_0, author = {L{\o}vgren, Alf Emil and Maday, Yvon and R{\o}nquist, Einar M.}, title = {A reduced basis element method for the steady {Stokes} problem}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {529--552}, publisher = {EDP-Sciences}, volume = {40}, number = {3}, year = {2006}, doi = {10.1051/m2an:2006021}, mrnumber = {2245320}, zbl = {1129.76036}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2006021/} }

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: Modélisation mathématique et analyse numérique PY - 2006 SP - 529 EP - 552 VL - 40 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2006021/ DO - 10.1051/m2an:2006021 LA - en ID - M2AN_2006__40_3_529_0 ER -

%0 Journal Article %A Løvgren, Alf Emil %A Maday, Yvon %A Rønquist, Einar M. %T A reduced basis element method for the steady Stokes problem %J ESAIM: Modélisation mathématique et analyse numérique %D 2006 %P 529-552 %V 40 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2006021/ %R 10.1051/m2an:2006021 %G en %F M2AN_2006__40_3_529_0

Løvgren, Alf Emil; Maday, Yvon; Rønquist, Einar M. A reduced basis element method for the steady Stokes problem. ESAIM: Modélisation mathématique et analyse numérique, Volume 40 (2006) no. 3, pp. 529-552. doi : 10.1051/m2an:2006021. http://www.numdam.org/articles/10.1051/m2an:2006021/

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