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

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 sense, then this CBS constant only depends on a pair of finite element spaces 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 H1, we investigate non-standard choices of H2 and the associated CBS constants, with the aim of designing efficient error estimators with effectivity indices close to one. When H1 and H2 satisfy certain conditions, we also prove new theoretical estimates for the CBS constant using linear algebra arguments.