TY - JOUR
T1 - Goal-oriented error estimation and adaptivity for elliptic PDEs with parametric or uncertain inputs
AU - Bespalov, Alex
AU - Praetorius, Dirk
AU - Rocchi, Leonardo
AU - Ruggeri, Michele
N1 - 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.
PY - 2019/3/1
Y1 - 2019/3/1
N2 - 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).
AB - 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).
KW - a posteriori error analysis
KW - finite element methods
KW - goal-oriented adaptivity
KW - parametric PDEs
KW - stochastic Galerkin methods
KW - two-level error estimate
UR - http://www.scopus.com/inward/record.url?scp=85058475419&partnerID=8YFLogxK
UR - https://arxiv.org/abs/1806.03928
U2 - 10.1016/j.cma.2018.10.041
DO - 10.1016/j.cma.2018.10.041
M3 - Article
AN - SCOPUS:85058475419
SN - 0045-7825
VL - 345
SP - 951
EP - 982
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
ER -