Finite element error analysis of elliptic PDES with random coefficients and its application to multilevel Monte Carlo methods

J. Charrier*, R. Scheichl, A. L. Teckentrup

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We consider a finite element approximation of elliptic partial differential equations with random coefficients. Such equations arise, for example, in uncertainty quantification in subsurface flow modeling. Models for random coefficients frequently used in these applications, such as log-normal random fields with exponential covariance, have only very limited spatial regularity and lead to variational problems that lack uniform coercivity and boundedness with respect to the random parameter. In our analysis we overcome these challenges by a careful treatment of the model problem almost surely in the random parameter, which then enables us to prove uniform bounds on the finite element error in standard Bochner spaces. These new bounds can then be used to perform a rigorous analysis of the multilevel Monte Carlo method for these elliptic problems that lack full regularity and uniform coercivity and boundedness. To conclude, we give some numerical results that confirm the new bounds.

Original languageEnglish
Pages (from-to)322-352
Number of pages31
JournalSiam journal on numerical analysis
Volume51
Issue number1
DOIs
Publication statusPublished - 2013

Keywords

  • Lack of full regularity
  • Log-normal coefficients
  • Not uniformly elliptic or bounded
  • PDEs with stochastic data
  • Truncated Karhunen-Loève expansion

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