Error estimates for semi-discrete finite element approximations for a moving boundary problem capturing the penetration of diffusants into rubber

TL;DR

This paper develops and analyzes semi-discrete finite element methods for a moving boundary problem modeling solvent diffusion into rubber, providing error estimates and numerical validation, laying groundwork for future mechanical modeling extensions.

Contribution

It introduces error estimates for semi-discrete finite element approximations of a moving boundary diffusion problem, combining energy estimates and numerical validation.

Findings

Error estimates for mass concentration and boundary position

Numerical results align with theoretical predictions

Method applicable to nonlinear parabolic moving boundary problems

Abstract

We consider a moving boundary problem with kinetic condition that describes the diffusion of solvent into rubber and study semi-discrete finite element approximations of the corresponding weak solutions. We report on both a priori and a posteriori error estimates for the mass concentration of the diffusants, and respectively, for the a priori unknown position of the moving boundary. Our working techniques include integral and energy-based estimates for a nonlinear parabolic problem posed in a transformed fixed domain combined with a suitable use of the interpolation-trace inequality to handle the interface terms. Numerical illustrations of our FEM approximations are within the experimental range and show good agreement with our theoretical investigation. This work is a preliminary investigation necessary before extending the current moving boundary modeling to account explicitly for the mechanics of hyperelastic rods to capture a directional swelling of the underlying elastomer.