Error analysis for a viscoelastic phase separation model

TL;DR

This paper introduces a stable finite element method for simulating viscoelastic phase separation, accurately capturing physical effects with proven error estimates and energy stability.

Contribution

It presents a novel unconditionally stable discretization that preserves energy structure and provides optimal error estimates for complex nonlinear viscoelastic models.

Findings

Method is unconditionally stable and energy-preserving.

Achieves order optimal error estimates.

Successfully reproduces physical effects in simulations.

Abstract

We consider 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 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.