CBS Constants & Their Role in Error Estimation for Stochastic Galerkin Finite Element Methods - Crowder, Adam J.; Powell, Catherine E. (Journal of scientific computing, 2018)
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 H 1 , H 2 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 , we investigate non-standard choices of H 2 and the associated CBS constants, with the aim of designing efficient error estimators with effectivity indices close to one. When H 1 and H 2 satisfy certain conditions, we also prove new theoretical estimates for the CBS constant using linear algebra arguments.