图书馆订阅: Guest
Begell Digital Portal Begell 数字图书馆 电子图书 期刊 参考文献及会议录 研究收集
国际不确定性的量化期刊
影响因子: 3.259 5年影响因子: 2.547 SJR: 0.531 SNIP: 0.8 CiteScore™: 1.52

ISSN 打印: 2152-5080
ISSN 在线: 2152-5099

Open Access

国际不确定性的量化期刊

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.


Articles with similar content:

A PRIORI ERROR ANALYSIS OF STOCHASTIC GALERKIN PROJECTION SCHEMES FOR RANDOMLY PARAMETRIZED ORDINARY DIFFERENTIAL EQUATIONS
International Journal for Uncertainty Quantification, Vol.6, 2016, issue 4
Christophe Audouze , Prasanth B. Nair
A HYBRID GENERALIZED POLYNOMIAL CHAOS METHOD FOR STOCHASTIC DYNAMICAL SYSTEMS
International Journal for Uncertainty Quantification, Vol.4, 2014, issue 1
Michael Schick, Vincent Heuveline
Numerical Method for Solving the Diffusive Lotke−Volterra Model with Discontinuous Coefficients for the Problem of Companies Competition
Journal of Automation and Information Sciences, Vol.44, 2012, issue 4
Andrey A. Yefimenko, Vitally V. Akimenko
Investigation of Parametric Iterative Processes for Solving Nonlinear Equations
Journal of Automation and Information Sciences, Vol.32, 2000, issue 1
Mikhail Ya. Bartish, Stepan M. Shakhno
Asymptotics of a Certain Optimal Problem with Two Singular Parameters
Journal of Automation and Information Sciences, Vol.45, 2013, issue 7
Vladimir G. Kozyrev