Goal-oriented error estimation and adaptivity for elliptic PDEs with parametric or uncertain inputs

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Bespalov, Alex and Praetorius, Dirk and Rocchi, Leonardo and Ruggeri, Michele (2019) Goal-oriented error estimation and adaptivity for elliptic PDEs with parametric or uncertain inputs. Computer Methods in Applied Mechanics and Engineering, 345. pp. 951-982. ISSN 0045-7825 (https://doi.org/10.1016/j.cma.2018.10.041)

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Abstract

We use the ideas of goal-oriented error estimation and adaptivity to design and implement an efficient adaptive algorithm for approximating linear quantities of interest derived from solutions to elliptic partial differential equations (PDEs) with parametric or uncertain inputs. In the algorithm, the stochastic Galerkin finite element method (sGFEM) is used to approximate the solutions to primal and dual problems that depend on a countably infinite number of uncertain parameters. Adaptive refinement is guided by an innovative strategy that combines the error reduction indicators computed for spatial and parametric components of the primal and dual solutions. The key theoretical ingredient is a novel two-level a posteriori estimate of the energy error in sGFEM approximations. We prove that this error estimate is reliable and efficient. The effectiveness of the goal-oriented error estimation strategy and the performance of the goal-oriented adaptive algorithm are tested numerically for three representative model problems with parametric coefficients and for three quantities of interest (including the approximation of pointwise values).

ORCID iDs

Bespalov, Alex, Praetorius, Dirk, Rocchi, Leonardo and Ruggeri, Michele ORCID logoORCID: https://orcid.org/0000-0001-6213-1602;
  • Item type:Article
    ID code:80552
    Dates:
    Date
    Event
    1 March 2019
    Published
    17 December 2018
    Published Online
    30 October 2018
    Accepted
    Notes:Funding Information: This work was initiated when AB visited the Institute for Analysis and Scientific Computing at TU Wien in 2017. AB would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme “Uncertainty quantification for complex systems: theory and methodologies”, where part of the work on this paper was undertaken. This work was supported by the EPSRC, United Kingdom , grant EP/K032208/1 . The work of AB and LR was supported by the EPSRC, United Kingdom under grant EP/P013791/1 . The work of DP and MR was supported by the Austrian Science Fund (FWF) under grants W1245 and F65 . Publisher Copyright: © 2018 Elsevier B.V.
    Subjects:Science > Mathematics
    Department:Faculty of Science > Mathematics and Statistics
    Depositing user: Pure Administrator
    Date deposited:05 May 2022 07:50
    Last modified:11 Nov 2024 13:28
    Related URLs:
    URI:https://strathprints.strath.ac.uk/id/eprint/80552
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