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International Journal for Uncertainty Quantification
IF: 0.967 5-Year IF: 1.301 SJR: 0.531 SNIP: 0.8 CiteScore™: 1.52

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2014007972
pages 423-454

SOME A PRIORI ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS OF ELLIPTIC AND PARABOLIC LINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

Christophe Audouze
University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Ontario, Canada M3H 5T6
Prasanth B. Nair
University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Ontario, Canada M3H 5T6

ABSTRACT

We study some theoretical aspects of Legendre polynomial chaos based finite element approximations of elliptic and parabolic linear stochastic partial differential equations (SPDEs) and provide a priori error estimates in tensor product Sobolev spaces that hold under appropriate regularity assumptions. Our analysis takes place in the setting of finite-dimensional noise, where the SPDE coefficients depend on a finite number of second-order random variables. We first derive a priori error estimates for finite element approximations of a class of linear elliptic SPDEs. Subsequently, we consider finite element approximations of parabolic SPDEs coupled with a Θ-weighted temporal discretization scheme. We establish conditions under which the time-stepping scheme is stable and derive a priori rates of convergence as a function of spatial, temporal, and stochastic discretization parameters. We later consider steady-state and time-dependent stochastic diffusion equations and illustrate how the general results provided here can be applied to specific SPDE models. Finally, we theoretically analyze primal and adjoint-based recovery of stochastic linear output functionals that depend on the solution of elliptic SPDEs and show that these schemes are superconvergent.