Energy norm a posteriori error estimation for parametric operator equations

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    Stochastic Galerkin approximation is an increasingly popular approach for the solution of elliptic PDE problems with correlated random data. A typical strategy is to combine conventional (h-)finite element approximation on the spatial domain with spectral (p-)approximation on a finite-dimensional manifold in the (stochastic) parameter domain. The issues involved in a posteriori error analysis of computed solutions are outlined in this paper using an abstract setting of parametric operator equations. A novel energy error estimator that uses a parameter-free part of the underlying differential operator is introduced which effectively exploits the tensor product structure of the approximation space. We prove that our error estimator is reliable and efficient. We also discuss different strategies for enriching the approximation space and prove two-sided estimates of the error reduction for the corresponding enhanced approximations. These give computable estimates of the error reduction that depend only on the problem data and the original approximation. © 2014 Society for Industrial and Applied Mathematics.
    Original languageEnglish
    Pages (from-to)A339-A363
    JournalSIAM Journal on Scientific Computing
    Issue number2
    Publication statusPublished - 13 Mar 2014


    • A posteriori error analysis
    • Error estimation
    • Karhunen-Loève expansion
    • Parametric operator equations
    • Random data
    • Stochastic finite elements
    • Stochastic Galerkin methods


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