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Computable error estimates for finite element approximations of elliptic partial differential equations with rough stochastic data
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.ORCID iD: 0000-0003-2669-359X
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
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2016 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 38, no 6, p. A3773-A3807Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics Publications , 2016. Vol. 38, no 6, p. A3773-A3807
Keywords [en]
A posteriori error, Elliptic PDE, Galerkin error, Lognormal, Monte Carlo methods, Quadrature error, Random PDE
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-201587DOI: 10.1137/15M1044266ISI: 000391853100029Scopus ID: 2-s2.0-85007124170OAI: oai:DiVA.org:kth-201587DiVA, id: diva2:1073443
Funder
Swedish Research Council, VR-621-2014-4776Swedish eā€Science Research Center
Note

QC 20170210

Available from: 2017-02-10 Created: 2017-02-10 Last updated: 2017-11-29Bibliographically approved

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  • apa
  • harvard1
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  • vancouver
  • Other style
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  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
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  • Other locale
More languages
Output format
  • html
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  • asciidoc
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