We analyze a posteriori error estimation and adaptive refinement algorithms for stochastic Galerkin Finite Element methods for countably-parametric, elliptic boundary value problems. A residual error estimator which separates the effects of gpc-Galerkin discretization in parameter space and of the Finite Element discretization in physical space in energy norm is established. It is proved that the adaptive algorithm converges. To this end, a contraction property of its iterates is proved. It is shown that the sequences of triangulations which are produced by the algorithm in the FE discretization of the active gpc coefficients are asymptotically optimal. Numerical experiments illustrate the theoretical results.
DOI : 10.1051/m2an/2015017
Mots-clés : Partial differential equations with random coefficients, generalized polynomial chaos, adaptive finite element methods, contraction property, residuala posteriori error estimation, uncertainty quantification
@article{M2AN_2015__49_5_1367_0,
author = {Eigel, Martin and Gittelson, Claude Jeffrey and Schwab, Christoph and Zander, Elmar},
title = {A convergent adaptive stochastic {Galerkin} finite element method with quasi-optimal spatial meshes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1367--1398},
publisher = {EDP-Sciences},
volume = {49},
number = {5},
year = {2015},
doi = {10.1051/m2an/2015017},
mrnumber = {3423228},
zbl = {1335.65006},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2015017/}
}
TY - JOUR AU - Eigel, Martin AU - Gittelson, Claude Jeffrey AU - Schwab, Christoph AU - Zander, Elmar TI - A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 1367 EP - 1398 VL - 49 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015017/ DO - 10.1051/m2an/2015017 LA - en ID - M2AN_2015__49_5_1367_0 ER -
%0 Journal Article %A Eigel, Martin %A Gittelson, Claude Jeffrey %A Schwab, Christoph %A Zander, Elmar %T A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 1367-1398 %V 49 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015017/ %R 10.1051/m2an/2015017 %G en %F M2AN_2015__49_5_1367_0
Eigel, Martin; Gittelson, Claude Jeffrey; Schwab, Christoph; Zander, Elmar. A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1367-1398. doi : 10.1051/m2an/2015017. http://www.numdam.org/articles/10.1051/m2an/2015017/
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