A Finite Element Framework for Bulk-Surface Coupled Pdes to Solve Moving Boundary Problems in Biophysics

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 ChengShen2022a. We show that mesh distortion is mitigated by adopting the pioneering work of DuanLi2024, 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.