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Solving high-dimensional random parametric PDEs poses a challengingcomputational problem. It is well-known that numerical methods can greatlybenefit from adaptive refinement algorithms, in particular when functionalapproximations in polynomials are computed as in stochastic Galerkin andstochastic collocations methods. This work investigates a residual basedadaptive algorithm used to approximate the solution of the stationary diffusionequation with lognormal coefficients. It is known that the refinement procedureis reliable, but the theoretical convergence of the scheme for this class ofunbounded coefficients remains a challenging open question. This paper advancesthe theoretical results by providing a quasi-error reduction results for theadaptive solution of the lognormal stationary diffusion problem. Acomputational example supports the theoretical statement.

Guaranteed quasi-error reduction of adaptive Galerkin FEM for parametric PDEs with lognormal coefficients