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Stochastic Galerkin finite element methods (SGFEMs) are commonly used to approximate solutions to PDEs with random inputs. However, the study of a posteriori error estimation strategies to drive adaptive enrichment of the associated tensor product spaces is still under development. In this work, we revisit an a posteriori error estimator introduced in Bespalov and Silvester (SIAM J Sci Comput 38(4):A2118–A2140, 2016) for SGFEM approximations of the parametric reformulation of the stochastic diffusion problem. A key issue is that the bound relating the true error to the estimated error involves a CBS (Cauchy–Buniakowskii–Schwarz) constant. If the approximation spaces associated with the parameter domain are orthogonal in a weighted $$L^2$$L2 sense, then this CBS constant only depends on a pair of finite element spaces $$H_{1}, H_{2}$$H1,H2 associated with the spatial domain and their compatibility with respect to an inner product associated with a parameter-free problem. For fixed choices of $$H_{1}$$H1, we investigate non-standard choices of $$H_{2}$$H2 and the associated CBS constants, with the aim of designing efficient error estimators with effectivity indices close to one. When $$H_1$$H1 and $$H_2$$H2 satisfy certain conditions, we also prove new theoretical estimates for the CBS constant using linear algebra arguments.

CBS Constants & Their Role in Error Estimation for Stochastic Galerkin Finite Element Methods