A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes^∗,∗∗

¹ Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany.
martin.eigel@wias-berlin.de
² Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA.
cgittels@purdue.edu
³ Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland.
schwab@sam.math.ethz.ch
⁴ Institute of Scientific Computing, Technical University Braunschweig, 38092 Braunschweig, Germany.
e.zander@tu-bs.de

Received: 27 December 2013
Revised: 25 February 2015

Abstract

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.