Fully-Discrete Finite Element Approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise: II. 2D and 3D Case
Provides rigorous error analysis for numerical solutions of a high-order stochastic PDE, which is important for computational mathematics but incremental in methodology.
The paper develops fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise in 2D and 3D, deriving strong a priori error estimates for both modeling and approximation errors.
We consider an initial- and Dirichlet boundary- value problem for a fourth-order linear stochastic parabolic equation, in two or three space dimensions, forced by an additive space-time white noise. Discretizing the space-time white noise a modeling 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 standard Galerkin finite element method based on C1 piecewise polynomials, and, for time-stepping, the Backward Euler method. We derive strong a priori estimates for the modeling error and for the approximation error to the solution of the regularized problem.