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Goal-Oriented Adaptivity for Multilevel Stochastic Galerkin FEM with Nonlinear Goal Functionals.

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Title: Goal-Oriented Adaptivity for Multilevel Stochastic Galerkin FEM with Nonlinear Goal Functionals.
Authors: Bespalov, Alex1 (AUTHOR) a.bespalov@bham.ac.uk, Praetorius, Dirk2 (AUTHOR) dirk.praetorius@asc.tuwien.ac.at, Ruggeri, Michele3 (AUTHOR) m.ruggeri@unibo.it
Source: SIAM Journal on Scientific Computing. 2026, Vol. 48 Issue 1, pA392-A417. 26p.
Subjects: Finite element method, Elliptic differential equations, Nonlinear functional analysis, Numerical calculations, Numerical analysis, Numerical solutions to partial differential equations, Error analysis in mathematics
Abstract: This paper is concerned with the numerical approximation of quantities of interest associated with solutions to parametric elliptic PDEs. The key novelty of this work is in its focus on the quantities of interest represented by continuously Gâteaux differentiable nonlinear functionals. We consider a class of parametric elliptic PDEs where the underlying differential operator has affine dependence on a countably infinite number of uncertain parameters. We design a goal-oriented adaptive algorithm for approximating nonlinear functionals of solutions to this class of parametric PDEs. In the algorithm, the approximations of parametric solutions to the primal and dual problems are computed using the multilevel stochastic Galerkin finite element method (SGFEM), and the adaptive refinement process is guided by reliable spatial and parametric error reduction indicators. We prove that the proposed algorithm generates multilevel SGFEM approximations for which the estimates of the error in the goal functional converge to zero. Numerical experiments for a selection of test problems and nonlinear quantities of interest illustrate and underpin our theoretical findings. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Scientific Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
An: 192099058
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  Data: Goal-Oriented Adaptivity for Multilevel Stochastic Galerkin FEM with Nonlinear Goal Functionals.
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  Data: <searchLink fieldCode="DE" term="%22Finite+element+method%22">Finite element method</searchLink><br /><searchLink fieldCode="DE" term="%22Elliptic+differential+equations%22">Elliptic differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Nonlinear+functional+analysis%22">Nonlinear functional analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+calculations%22">Numerical calculations</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+solutions+to+partial+differential+equations%22">Numerical solutions to partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Error+analysis+in+mathematics%22">Error analysis in mathematics</searchLink>
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  Data: This paper is concerned with the numerical approximation of quantities of interest associated with solutions to parametric elliptic PDEs. The key novelty of this work is in its focus on the quantities of interest represented by continuously Gâteaux differentiable nonlinear functionals. We consider a class of parametric elliptic PDEs where the underlying differential operator has affine dependence on a countably infinite number of uncertain parameters. We design a goal-oriented adaptive algorithm for approximating nonlinear functionals of solutions to this class of parametric PDEs. In the algorithm, the approximations of parametric solutions to the primal and dual problems are computed using the multilevel stochastic Galerkin finite element method (SGFEM), and the adaptive refinement process is guided by reliable spatial and parametric error reduction indicators. We prove that the proposed algorithm generates multilevel SGFEM approximations for which the estimates of the error in the goal functional converge to zero. Numerical experiments for a selection of test problems and nonlinear quantities of interest illustrate and underpin our theoretical findings. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of SIAM Journal on Scientific Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1137/23M1597678
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      – Code: eng
        Text: English
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      Pagination:
        PageCount: 26
        StartPage: A392
    Subjects:
      – SubjectFull: Finite element method
        Type: general
      – SubjectFull: Elliptic differential equations
        Type: general
      – SubjectFull: Nonlinear functional analysis
        Type: general
      – SubjectFull: Numerical calculations
        Type: general
      – SubjectFull: Numerical analysis
        Type: general
      – SubjectFull: Numerical solutions to partial differential equations
        Type: general
      – SubjectFull: Error analysis in mathematics
        Type: general
    Titles:
      – TitleFull: Goal-Oriented Adaptivity for Multilevel Stochastic Galerkin FEM with Nonlinear Goal Functionals.
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            NameFull: Bespalov, Alex
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            NameFull: Praetorius, Dirk
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            NameFull: Ruggeri, Michele
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            – D: 01
              M: 01
              Text: 2026
              Type: published
              Y: 2026
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            – TitleFull: SIAM Journal on Scientific Computing
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