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

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 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).

Numerical analysis for time-dependent advection-diffusion problems with random discontinuous coefficients

“…This generalizes the elliptic setting which has drawn great attention over the last decades. While many publications focus on numerical methods for continuous stochastic coefficients (see, e.g., [1,[4][5][6][7]12,16,17,23,29,33,38,39,43,45,46]), the literature on stochastic discontinuous coefficients or stochastic interface problems is sparse (see, e.g., [28,32,47]). The reasons are manifold: Gaussian random fields are well-defined mathematical objects and their properties are well studied, simulation methods range from spectral approximations to Fourier methods (see, e.g., [25,31,44]).…”

Numerical analysis for time-dependent advection-diffusion problems with random discontinuous coefficients

“…This work is a generalization to the elliptic setting which has drawn attention over the last decades. While many publications focus on numerical methods for continuous stochastic coefficients (see, e.g., [1,3,4,5,6,9,16,17,23,30,34,35,36,40,42,44]), the literature on stochastic discontinuous coefficients or stochastic interface problems is sparse (see, e.g., [29,33,45]). The reasons are twofold: On one hand a Gaussian random field is a well defined mathematical object and its properties are well studied, on the other hand there is no general definition and approximation method for a (discontinuous) Lévy field.…”