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    Computable Error Estimates for Finite Element Approximations of Elliptic Partial Differential Equations with Rough Stochastic Data

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    Type
    Article
    Authors
    Hall, Eric Joseph
    Hoel, Håkon
    Sandberg, Mattias
    Szepessy, Anders
    Tempone, Raul cc
    KAUST Department
    Applied Mathematics and Computational Science Program
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    KAUST Grant Number
    CRG3 Award ref. 2281
    Date
    2016-12-08
    Online Publication Date
    2016-12-08
    Print Publication Date
    2016-01
    Permanent link to this record
    http://hdl.handle.net/10754/622637
    
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    Abstract
    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.
    Citation
    Hall 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.
    Sponsors
    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.
    Publisher
    Society for Industrial & Applied Mathematics (SIAM)
    Journal
    SIAM Journal on Scientific Computing
    DOI
    10.1137/15M1044266
    arXiv
    1510.02708
    Additional Links
    http://epubs.siam.org/doi/10.1137/15M1044266
    ae974a485f413a2113503eed53cd6c53
    10.1137/15M1044266
    Scopus Count
    Collections
    Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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