An implementation of a stochastic partial differential equation in FEniCS

Abstract

Computational platforms based on intuitive and efficient software can allow for a easy and accessible computational mathematical modelling. FEniCS is an opensource software that allows for automated way of implementing finite element method(FEM) code for partial differential equations from the variational formulation of a differential equation. Using the FEniCS platform we implement the Poisson equation with variable coefficients r · (a(x)ru(x)) = f(x). The noise term is chosen as a = exp(T) where T is a Gaussian random field modelled as a solution to a stochastic differential equation of the form (− ) /2T = W 2 N. where W. This equation was solved using fractional approximation of the operator and a finite element discretisation. For both equations a thorough step-by-step implementation is presented with the aforementioned equation being completely implemented in FEniCS. As a measure of quality of implementation a strong mean square error was estimated. The convergence rate of the strong error results for the Gaussian random field is compared to the theoretical results. The result of our implementation quality, confirms the theory.