RT Journal Article
ID 207aeb651cd2b5fd
A1 Audouze , Christophe
A1 Nair, Prasanth B.
T1 SOME A PRIORI ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS OF ELLIPTIC AND PARABOLIC LINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
JF International Journal for Uncertainty Quantification
JO IJUQ
YR 2014
FD 2014-08-29
VO 4
IS 5
SP 423
OP 454
K1 stochastic partial differential equations
K1 a priori error estimation
K1 chaos expansions
K1 finite element methods
K1 time-stepping stability
K1 functional approximation
AB 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.
PB Begell House
LK http://dl.begellhouse.com/journals/52034eb04b657aea,2f7b99cc281f2702,207aeb651cd2b5fd.html