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Error analysis for a viscoelastic phase separation model

Aaron Brunk, Herbert Egger, Oliver Habrich, Maria Lukacova-Medvidova

TL;DR

The paper tackles numerically simulating viscoelastic phase separation in polymer-solvent mixtures described by a coupled parabolic-hyperbolic system that combines a Cahn-Hilliard-type equation with viscoelastic stress. It introduces a fully discrete, structure-preserving finite element scheme using a variational time discretization, which exactly preserves the energy-dissipation relation and mass conservation. The authors prove existence and, under a mild $h$–$\tau$ compatibility condition, uniqueness of discrete solutions, and establish order-optimal error estimates of $O(h^4+\tau^4)$ via a relative-energy stability framework. Numerical experiments corroborate the theoretical convergence rates and reproduce the characteristic multi-stage viscoelastic phase separation dynamics observed in experiments, validating the method’s accuracy and physical relevance.

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.

Error analysis for a viscoelastic phase separation model

TL;DR

The paper tackles numerically simulating viscoelastic phase separation in polymer-solvent mixtures described by a coupled parabolic-hyperbolic system that combines a Cahn-Hilliard-type equation with viscoelastic stress. It introduces a fully discrete, structure-preserving finite element scheme using a variational time discretization, which exactly preserves the energy-dissipation relation and mass conservation. The authors prove existence and, under a mild – compatibility condition, uniqueness of discrete solutions, and establish order-optimal error estimates of via a relative-energy stability framework. Numerical experiments corroborate the theoretical convergence rates and reproduce the characteristic multi-stage viscoelastic phase separation dynamics observed in experiments, validating the method’s accuracy and physical relevance.

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.
Paper Structure (17 sections, 13 theorems, 109 equations, 1 figure, 2 tables)

This paper contains 17 sections, 13 theorems, 109 equations, 1 figure, 2 tables.

Table of Contents

  1. Notation, assumptions, and preliminaries
  2. Proposed method and main results
  3. Proof of Theorem \ref{['thm:ex']}
  4. A discrete stability estimate
  5. Error estimates
  6. Uniqueness of discrete solutions
  7. Numerical illustration
  8. Convergence rates
  9. Qualitative behavior
  10. Conclusion & Outlook
  11. Projection error estimates
  12. Space discretization
  13. Discrete interpolation
  14. Time discretization
  15. Projection estimates for nonlinear terms
  16. ...and 2 more sections

Key Result

Theorem 3

Let (A0)--(A7) hold. Then for any $\phi_{h,0},q_{h,0} \in \mathcal{V}_h$, Problem prob:full has at least one solution. Moreover, any such solution conserves mass and dissipates energy, i.e., for all $0 \le m \le n \le N$. Here the energy and dissipation functionals $\mathcal{E}(\phi,q)$, $\mathcal{D}_{\phi}(\mu,q)$ are defined after eq:energy. Furthermore, solutions can be uniformly bounded by T

Figures (1)

  • Figure 1: Snapshots of the volume fraction $\phi$ obtained for the second test case, illustrating typical stages of viscoelastic phase separation.

Theorems & Definitions (17)

  • Theorem 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • proof
  • Lemma 7
  • Lemma 8
  • proof
  • Lemma 9
  • Lemma 10
  • ...and 7 more