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

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...

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Bibliographic Details
Published in:SIAM journal on scientific computing 2016-01, Vol.38 (6), p.A3773-A3807
Main Authors: Hall, Eric Joseph, Hoel, Håkon, Sandberg, Mattias, Szepessy, Anders, Tempone, Raúl
Format: Article
Language:English
Subjects:
A posteriori error
Elliptic PDE
Galerkin error
Lognormal
Monte Carlo methods
Quadrature error
Random PDE
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Summary: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.
ISSN:1064-8275
1095-7197
1095-7197
DOI:10.1137/15M1044266