Error analysis for a second order approximation of a viscoelastic phase separation model

Abstract

We consider a systematic numerical approximation of a viscoelastic phase separation model that describes the demixing of a polymer-solvent mixture. An unconditionally stable discretisation method is proposed based on a finite element approximation in space and a variational time discretization strategy. The proposed method preserves the energy-dissipation structure of the underlying system exactly and allows us to establish a fully discrete nonlinear stability estimate in natural norms based on the concept of relative energy. These estimates are used to derive order optimal error estimates for the method under minimal smoothness assumptions on the problem data, despite the presence of various strong nonlinearities in the equations. The theoretical results and main properties of the method are illustrated by numerical simulations, which also demonstrate the capability to reproduce the relevant physical effects observed in experiments.