Generalized polynomial chaos decomposition and spectral methods for the stochastic Stokes equations

Dong, Qingshun; Xu, Chuanju

doi:10.1016/j.compfluid.2012.10.001

Cited by 3 publications

(2 citation statements)

References 31 publications

(36 reference statements)

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“…Mugler and Starkloff [17] proposed a new approach based on a stochastic Petrov-Galerkin projection scheme for solving the steady-state stochastic diffusion equation with boundedness assumptions on the random coefficients weaker than those usually considered in the literature. Qu and Xu [18] presented convergence analysis of a stochastic Galerkin approach for solving the Stokes equations with random coefficients, whose solution is discretized using spectral and generalized polynomial chaos (PC) expansions for its spatial and random part, respectively. In particular, the analysis of Babuska et al [7] for stochastic elliptic SPDEs is extended to saddle-point problems.…”

Section: Nomenclaturementioning

confidence: 99%

Some a Priori Error Estimates for Finite Element Approximations of Elliptic and Parabolic Linear Stochastic Partial Differential Equations

Audouze

Nair

2014

Int. J. UncertaintyQuantification

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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 finitedimensional 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. KEY WORDS: stochastic partial differential equations, a priori error estimation, chaos expansions, finite element methods, time-stepping stability, functional approximation * Correspond to Prasanth B. Nair,We shall now introduce the finite-dimensional subspaces V h ⊂ V and S p ξ ⊂ S used for the numerical approximation of (4) and (9). First, let T be a triangulation of the domain D consisting of a finite collection of triangles (resp. tetrahedra) T i such that T i ∩ T j = ∅ for i = j, i T i = D, and such that no vertex lies in the interior of an edge (resp. a face) of another triangle (resp. tetrahedron). We consider a family of triangulations T h with mesh-size h ∈ [0, 1[, which are supposed to be nondegenerate, i.e., there exists a constant µ > 0 such that diam(B T ) ≥ µ diam(T ), for all T ∈ T h and h ∈ [0, 1[, where B T is the largest ball contained in T , and such that max diam(T ), T ∈ T h ≤ h diam(D).

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Section: Nomenclaturementioning

confidence: 99%

Some a Priori Error Estimates for Finite Element Approximations of Elliptic and Parabolic Linear Stochastic Partial Differential Equations

Audouze

Nair

2014

Int. J. UncertaintyQuantification

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“…In recent years, the PC method has been applied to a wide spectrum of stochastic problems, for instance to the theory of elasticity and plasticity (Ghamen, Spanos 1991;Anders, Hori 1999), stochastic dynamics (Ghanem, Spanos 1993;Sarkar, Ghanem 2002), wave propagation in random media (Manolis, Karakostas 2003), stochastic Stokes equations (Qu, Xu 2013) or sensitivity analysis (Sudret 2008;Crestaux et al 2009;Blatman, Sudret 2010). In the mentioned examples, the application of the PC expansion presents the evidence of a good calculation efficiency.…”

Section: Polynomial Chaos Approximation -Theoretical Background Litementioning

confidence: 99%

A Conditional Stochastic Projection Method Applied to a Parametric Vibrations Problem

Brząkała

Herbut

2014

Journal of Civil Engineering and Management

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Abstract. Parametric vibrations can be observed in cable-stayed bridges due to periodic excitations caused by a deck or a pylon. The vibrations are described by an ordinary differential equation with periodic coefficients. The paper focuses on random excitations, i.e. on the excitation amplitude and the excitation frequency which are two random variables. The excitation frequency L ω is discretized to a finite sequence of representative points,ω . Therefore, the problem is (conditionally) formulated and solved as a one-dimensional polynomial chaos expansion generated by the random excitation amplitude. The presented numerical analysis is focused on a real situation for which the problem of parametric resonance was observed (a cable of the Ben-Ahin bridge). The results obtained by the use of the conditional polynomial chaos approximations are compared with the ones based on the Monte Carlo simulation (truly two-dimensional, not conditional one). The convergence of both methods is discussed. It is found that the conditional polynomial chaos can yield a better convergence then the Monte Carlo simulation, especially if resonant vibrations are probable.

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Symplectic Approach on the Wave Propagation Problem for Periodic Structures with Uncertainty

Zhao

Liang

Zhang

et al. 2019

Acta Mech. Solida Sin.

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Generalized polynomial chaos decomposition and spectral methods for the stochastic Stokes equations

Cited by 3 publications

References 31 publications

Some a Priori Error Estimates for Finite Element Approximations of Elliptic and Parabolic Linear Stochastic Partial Differential Equations

Some a Priori Error Estimates for Finite Element Approximations of Elliptic and Parabolic Linear Stochastic Partial Differential Equations

A Conditional Stochastic Projection Method Applied to a Parametric Vibrations Problem

Symplectic Approach on the Wave Propagation Problem for Periodic Structures with Uncertainty

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