We derive computable error estimates for finite element approximations of linear elliptic partial differential equations with rough stochastic coefficients. In this setting, the exact solutions contain high frequency content that standard a posteriori error estimates fail to capture. We propose goal-oriented estimates, based on local error indicators, for the pathwise Galerkin and expected quadrature errors committed in standard, continuous, piecewise linear finite element approximations. Derived using easily validated assumptions, these novel estimates can be computed at a relatively low cost and have applications to subsurface flow problems in geophysics where the conductivities are assumed to have lognormal distributions with low regularity. Our theory is supported by numerical experiments on test problems in one and two dimensions.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): CRG3 Award ref. 2281
Acknowledgements: This research was supported by Swedish Research Council grant VR-621-2014-4776 and the Swedish e-Science Research Center. It was carried out while the first author was a Goran Gustafsson postdoctoral fellow at KTH Royal Institute of Technology. The second author was supported by Norges Forskningsrad, research project 214495 LIQCRY. The fifth author is a member of the KAUST Strategic Research Initiative, Center for Uncertainty Quantification in Computational Sciences and Engineering, and was supported by the KAUST CRG3 Award ref. 2281.