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

Abstract

We consider a model 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. First, we approximate its solution by the solution of an auxiliary fourth order stochastic parabolic problem with additive, finite dimensional, spectral-type stochastic load. Then, fully discrete approximations of the solution to the approximate problem are constructed by using, for the discretization in space, a standard Galerkin finite element method based on $H^2$-piecewise polynomials and, for time-stepping, the Crank--Nicolson method. Analyzing the proposed discretization approach, we derive strong error estimates that establish optimal rate of convergence without imposing CFL conditions on the discretization parameters. In contrast to the backward Euler method, the Crank--Nicolson method is not strongly $A(\vartheta)$-stable, and thus a different stability argument is required for building up the convergence analysis.