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

Handle URI:
http://hdl.handle.net/10754/622637
Title:
Computable Error Estimates for Finite Element Approximations of Elliptic Partial Differential Equations with Rough Stochastic Data
Authors:
Hall, Eric Joseph; Hoel, Håkon; Sandberg, Mattias; Szepessy, Anders; Tempone, Raul ( 0000-0003-1967-4446 )
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.
KAUST Department:
Applied Mathematics and Computational Science Program
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.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Scientific Computing
KAUST Grant Number:
CRG3 Award ref. 2281
Issue Date:
8-Dec-2016
DOI:
10.1137/15M1044266
Type:
Article
ISSN:
1064-8275; 1095-7197
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.
Additional Links:
http://epubs.siam.org/doi/10.1137/15M1044266
Appears in Collections:
Articles; Applied Mathematics and Computational Science Program

Full metadata record

DC FieldValue Language
dc.contributor.authorHall, Eric Josephen
dc.contributor.authorHoel, Håkonen
dc.contributor.authorSandberg, Mattiasen
dc.contributor.authorSzepessy, Andersen
dc.contributor.authorTempone, Raulen
dc.date.accessioned2017-01-08T13:05:38Z-
dc.date.available2017-01-08T13:05:38Z-
dc.date.issued2016-12-08en
dc.identifier.citationHall 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.en
dc.identifier.issn1064-8275en
dc.identifier.issn1095-7197en
dc.identifier.doi10.1137/15M1044266en
dc.identifier.urihttp://hdl.handle.net/10754/622637-
dc.description.abstractWe 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.en
dc.description.sponsorshipThis 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.en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.relation.urlhttp://epubs.siam.org/doi/10.1137/15M1044266en
dc.rightsArchived with thanks to SIAM Journal on Scientific Computingen
dc.subjectA posteriori erroren
dc.subjectElliptic PDEen
dc.subjectGalerkin erroren
dc.subjectLognormalen
dc.subjectMonte Carlo methodsen
dc.subjectQuadrature erroren
dc.subjectRandom PDEen
dc.titleComputable Error Estimates for Finite Element Approximations of Elliptic Partial Differential Equations with Rough Stochastic Dataen
dc.typeArticleen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.identifier.journalSIAM Journal on Scientific Computingen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionDepartment of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA, 10030, United Statesen
dc.contributor.institutionDepartment of Mathematics, University of Oslo, Oslo, 0316, Norwayen
dc.contributor.institutionDepartment of Mathematics, KTH Royal Institute of Technology, Stockholm, 100 44, Swedenen
kaust.authorTempone, Raulen
kaust.grant.numberCRG3 Award ref. 2281en
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