CBS Constants and 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 in its infancy. In this work, we revisit an error estimator introduced in [A. Bespalov, D. Silvester, Efficient adaptive stochastic Galerkin methods for parametric operator equations, SIAM J. Sci. Comput., 38(4), 2016] for SGFEM approximations of the parametric reformulation of the stochastic diffusion problem. We show that the proven bound for that error estimator can be derived using classical theory that is well known for determinstic Galerkin FEMs. A key issue is that the bound involves a CBS (Cauchy-Buniakowskii-Schwarz) constant. If the approximation spaces associated with the parameter domain are orthogonal, then this CBS constant only depends on a pair of finite element spaces H1, H2 and their compatibility with respect to an inner product associated with a parameter-free problem. For fixed choices of H2, we investigate non-standard choices of H2 and the associated CBS constants, with the aim of designing efficient error estimators. When H1 and H2 satisfy certain conditions, we also prove theoretical estimates for the CBS constant using a novel linear algebra approach. Our results are also applicable to the design of adaptive finite element schemes for deterministic PDEs