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A Finite Element framework for bulk-surface coupled PDEs to solve moving boundary problems in biophysics

Alessandro Contri, André Massing, Padmini Rangamani

TL;DR

The paper introduces a finite element framework for solving bulk–surface PDEs on moving boundaries relevant to biophysics, unifying structure-preserving discretization with ALE-based mesh control. It extends a proven bound/mass preservation approach to moving domains, adds tangential mesh redistribution via a two-stage ALE map, and employs CIP stabilization to handle advection-dominated regimes on evolving interfaces. The authors demonstrate convergence and stability across ADR, Cahn–Hilliard, and Helfrich/Willmore models, including coupled bulk–surface systems and benchmark tests such as Kovács–Li–Lubich and tumor-growth-like scenarios. The resulting methodology is flexible, post-processing-based, and scalable to complex membrane reshaping problems encountered in cellular biology.

Abstract

We consider moving boundary problems for biophysics and introduce a new computational framework to handle the complexity of the bulk-surface PDEs. In our framework, interpretability is maintained by adapting the fast, generalizable and accurate structure preservation scheme in [Q. Cheng and J. Shen, \textit{Computer Methods in Applied Mechanics and Engineering}, 391 (2022)]. We show that mesh distortion is mitigated by adopting the pioneering work of [B. Duan and B. Li, \textit{SIAM J. Sci. Comput.}, 46 (2024)], which is tied to an Arbitrary Lagrangian Eulerian (ALE) framework. We test our algorithms accuracy on moving surfaces with boundary for the following PDEs: advection-diffusion-reaction equations, phase-field models of Cahn-Hilliard type, and Helfrich energy gradient flows. We performed convergence studies for all the schemes introduced to demonstrate accuracy. We use a staggered approach to achieve coupling and further verify the convergence of this coupling using numerical experiments. Finally, we demonstrate broad applicability of our work by simulating state-of-the-art tests of biophysical models that involve membrane deformation.

A Finite Element framework for bulk-surface coupled PDEs to solve moving boundary problems in biophysics

TL;DR

The paper introduces a finite element framework for solving bulk–surface PDEs on moving boundaries relevant to biophysics, unifying structure-preserving discretization with ALE-based mesh control. It extends a proven bound/mass preservation approach to moving domains, adds tangential mesh redistribution via a two-stage ALE map, and employs CIP stabilization to handle advection-dominated regimes on evolving interfaces. The authors demonstrate convergence and stability across ADR, Cahn–Hilliard, and Helfrich/Willmore models, including coupled bulk–surface systems and benchmark tests such as Kovács–Li–Lubich and tumor-growth-like scenarios. The resulting methodology is flexible, post-processing-based, and scalable to complex membrane reshaping problems encountered in cellular biology.

Abstract

We consider moving boundary problems for biophysics and introduce a new computational framework to handle the complexity of the bulk-surface PDEs. In our framework, interpretability is maintained by adapting the fast, generalizable and accurate structure preservation scheme in [Q. Cheng and J. Shen, \textit{Computer Methods in Applied Mechanics and Engineering}, 391 (2022)]. We show that mesh distortion is mitigated by adopting the pioneering work of [B. Duan and B. Li, \textit{SIAM J. Sci. Comput.}, 46 (2024)], which is tied to an Arbitrary Lagrangian Eulerian (ALE) framework. We test our algorithms accuracy on moving surfaces with boundary for the following PDEs: advection-diffusion-reaction equations, phase-field models of Cahn-Hilliard type, and Helfrich energy gradient flows. We performed convergence studies for all the schemes introduced to demonstrate accuracy. We use a staggered approach to achieve coupling and further verify the convergence of this coupling using numerical experiments. Finally, we demonstrate broad applicability of our work by simulating state-of-the-art tests of biophysical models that involve membrane deformation.

Paper Structure

This paper contains 22 sections, 75 equations, 10 figures.

Table of Contents

  1. Introduction
  2. State of the art
  3. New contributions and outline
  4. Model
  5. Space discretization
  6. Time discretization
  7. Structure preservation
  8. Tangential grid control and ALE discretization
  9. Advection-diffusion-reaction system
  10. Numerical methods for advection-dominant problems
  11. Numerical results for the ADR system
  12. Cahn-Hilliard system as a phase separation model
  13. Numerical methods for Cahn-Hilliard system
  14. Numerical results for Cahn-Hilliard system
  15. Membrane force system
  16. ...and 7 more sections

Figures (10)

  • Figure 1: Sketch of an open and closed geometry for notation purposes. One is a 3D object and the other is a 2D object.
  • Figure 1: Convergence studies for the solver adrSolver 4 in List \ref{['item:ad_solvers']}. As expected, first and second order convergence are achieved in time and space, respectively.
  • Figure 1: Experimental studies for potential $F_1(u)=\frac{1}{4}((1-u) \log(1-u) + (1 + u) \log(1 + u)) + \frac{1-u^2}{2}$ and the different solvers in List \ref{['item:ch_solvers']}.
  • Figure 1: Convergence studies for Willmore flow of sphere under spontaneous mean curvature for the solvers in List \ref{['item:willmore_solvers']}.
  • Figure 1: Convergence studies for mcadrSolver 4 and problem \ref{['eq:mc_adr_conv_kovacs']}.
  • ...and 5 more figures

Theorems & Definitions (8)

  • Remark 2.1
  • Remark 3.1
  • Remark 6.1
  • Remark 7.1
  • Remark 7.2
  • Remark 7.3
  • Remark 7.4
  • Remark 8.1