Numerical schemes for a moving boundary problem describing the penetration of diffusants into rubber

We present two numerical schemes for a moving-boundary problem that describes the penetration of diffusants into rubber.  Our goal is to approximate the penetration front (i.e., the position of the moving boundary) and the diffusant profile.  Our first numerical scheme utilizes the Galerkin finite element method for space discretization and the backward Euler method for time discretization. We analyze both semi-discrete and fully discrete approximations of the weak solution to the model equations. We prove a priori and a posteriori error estimates, and present numerical simulation results. As an alternative approach to finite element approximation, we introduce a random walk algorithm that employs a finite number of particles to approximate both the diffusant profile and the location of the penetration front. The transport of diffusants is due to unbiased randomness, while the evolution of the penetration front is based on biased randomness.  Simulation results obtained via the random walk approach are comparable with the one based on the finite element method.

At the end, I will briefly discuss about FEniCS and provide an example of solving PDE using FEniCS.