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A. Bespalov, D. Praetorius, L. Rocchi, M. Ruggeri:
"Convergence of adaptive stochastic Galerkin FEM";
in: "ASC Report 31/2018", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2018, ISBN: 978-3-902627-11-7, 1 - 28.

We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty.Four different marking strategies are employed for refinement of stochastic Galerkin finite element approximations.
The algorithms are driven by the energy error reduction estimates derived from two-level a posteriori error indicators for spatial approximations and hierarchical a posteriori error indicators for parametric approximations. The focus of this work is on the mathematical foundation of the adaptive algorithms in the sense of rigorous convergence analysis. In particular, we prove that the proposed algorithms drive the underlying energy error estimates to zero.

adaptive methods, a posteriori error analysis, convergence, two-level errorestimate, stochastic Galerkin methods, finite element methods, parametric PDEs.

http://www.asc.tuwien.ac.at/preprint/2018/asc31x2018.pdf