Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise

Abstract

Abstract.We consider an initial and Dirichlet boundary value problem for a fourth-order linear stochastic parabolic equation, in one space dimension, forced by an additive space-time white noise. Discretizing the space-time white noise a modelling error is introduced and a regularized fourth-order linear stochastic parabolic problem is obtained. Fully-discrete approximations to the solution of the regularized problem are constructed by using, for discretization in space, a Galerkin finite element method based on C 0 or C 1 piecewise polynomials, and, for time-stepping, the Backward Euler method. We derive strong a priori estimates for the modelling error and for the approximation error to the solution of the regularized problem.Mathematics Subject Classification. 65M60, 65M15, 65C20.

Crank--Nicolson Finite Element Approximations for a Linear Stochastic Fourth Order Equation with Additive Space-Time White Noise

“…. (4.10) 11 We continue by estimating the terms in the right hand side of (4.9) and (4.10). First, we observe that v(τ 1 , ·) − v(τ 1 2 , ·) 2 0,D ≤ ∆τ 2 τ1 τ 1 2 ∂ τ v(τ, ·) 2 0,D dτ , which, along with (2.16) (with ℓ = 1, p = 0, β = 0), yields…”

“…We will get the error estimate (4.38) by interpolation after proving it for θ = 1 and θ = 0 (cf. [12]). In the sequel, we will use the symbol C to denote a generic constant that is independent of ∆τ and h, and may change value from one line to the other.…”

Finite-element approximation of the linearized Cahn-Hilliard-Cook equation

“…There are few studies of numerical methods for the Cahn-Hilliard-Cook equation. We are only aware of [4] in which convergence in probability was proved for a difference scheme for the nonlinear equation in multiple dimensions, and [10] where strong convergence was proved for the finite element method for the linear equation in 1-D.…”

Finite Element Approximation of the Cahn–Hilliard–Cook Equation

“…We are only aware of [3] in which convergence in probability was proved for a difference scheme for the nonlinear equation in multiple dimensions. For the linear equation there is [10], where strong convergence estimates were proved for the finite element method for the linear equation in 1-D, and the already mentioned work [12] on the finite element method for the stochastic convolution in multiple dimensions.…”