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

• Published in: Journal of scientific computing Vol. 77; no. 2; pp. 1030 - 1054
• Main Authors: Crowder, Adam J., Powell, Catherine E.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.11.2018; Springer Nature B.V
• Subjects: Algorithms; Approximation; Computational Mathematics and Numerical Analysis; Error analysis; Finite element method; Galerkin method; Hilbert space; Investigations; Linear algebra; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Parameters; Partial differential equations; Random variables; Tensors; Theoretical; 35R60; CBS constants; Error estimation for stochastic PDE problems; 65F10; Stochastic finite element method; Stochastic Galerkin method; 65N30; Strengthened Cauchy–Schwarz inequality; 60H35
• ISSN: 0885-7474; 1573-7691; 1573-7691
• DOI: 10.1007/s10915-018-0736-4

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.