Partial differential equations (PDEs) with inputs that depend on infinitely many parameters pose serious theoretical and computational challenges. Sophisticated numerical algorithms that automatically determine which parameters need to be activated in the approximation space in order to estimate a quantity of interest to a prescribed error tolerance are needed. For elliptic PDEs with parameter-dependent coefficients, stochastic Galerkin finite element methods (SGFEMs) have been well studied. Under certain assumptions, it can be shown that there exists a sequence of SGFEM approximation spaces for which the energy norm of the error decays to zero at a rate that is independent of the number of input parameters. However, it is not clear how to adaptively construct these spaces in a practical and computationally efficient way. We present a new adaptive SGFEM algorithm that tackles elliptic PDEs with parameter-dependent coefficients quickly and efficiently. We consider approximation spaces with a multilevel structure-where each solution mode is associated with a finite element space on a potentially different mesh-and use an implicit a posteriori error estimation strategy to steer the adaptive enrichment of the space. At each step, the components of the error estimator are used to assess the potential benefits of a variety of enrichment strategies, including whether or not to activate more parameters. No marking or tuning parameters are required. Numerical experiments for a selection of test problems demonstrate that the new method performs optimally in that it generates a sequence of approximations for which the estimated energy error decays to zero at the same rate as the error for the underlying finite element method applied to the associated parameter-free problem.

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.

No abstract