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A. Bespalov, D. Praetorius, M. Ruggeri:
"Convergence and rate optimality of adaptive multilevel stochastic Galerkin FEM";
IMA J. Numer. Anal., accepted for publication (2021).

We analyze an adaptive algorithm for the numerical solution of parametric elliptic partial differential equations in two-dimensional physical domains, with coefficients and right-hand side functions depending on infinitely many (stochastic) parameters. The algorithm generates multilevel stochastic Galerkin approximations; these are represented in terms of a sparse generalized polynomial chaos expansion with coefficients residing in finite element spaces associated with different locally refined meshes. Adaptivity is driven by a two-level a posteriori error estimator and employs a Dörfler-type marking on the joint set of spatial and parametric error indicators. We show that, under an appropriate saturation assumption, the proposed adaptive strategy yields optimal convergence rates with respect to the overall dimension of the
underlying multilevel approximation spaces.

adaptive methods; a posteriori error analysis; two-level error estimation; multilevel stochastic Galerkin method; finite element methods; parametric PDEs

http://dx.doi.org/10.1093/imanum/drab036