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

Eric Joseph Hall, Håkon Hoel, Mattias Sandberg, Anders Szepessy, Raúl Tempone

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)A3773–A3807
Number of pages35
JournalSIAM Journal on Scientific Computing
Volume38
Issue number6
Early online date8 Dec 2016
DOIs
Publication statusPublished - 2016

Keywords

  • Random PDE
  • Monte Carlo Method
  • A posteriori error analysis
  • Galerkin
  • Quadrature
  • Elliptic PDE
  • Lognormal
  • Rough stochastic data

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