Computable Error Estimates for Finite Element Approximations of Elliptic Partial Differential Equations with Rough Stochastic Data
KAUST DepartmentApplied Mathematics and Computational Science Program
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
KAUST Grant NumberCRG3 Award ref. 2281
Online Publication Date2016-12-08
Print Publication Date2016-01
Permanent link to this recordhttp://hdl.handle.net/10754/622637
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AbstractWe 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.
CitationHall EJ, Hoel H, Sandberg M, Szepessy A, Tempone R (2016) Computable Error Estimates for Finite Element Approximations of Elliptic Partial Differential Equations with Rough Stochastic Data. SIAM Journal on Scientific Computing 38: A3773–A3807. Available: http://dx.doi.org/10.1137/15M1044266.
SponsorsThis 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.