A finite element framework for solving coupled multiphysics problem with moving boundaries in cell biophysics

Cellular morphodynamics requires solving systems of coupled partial differential equations on moving bulk and surface domains, where advection-dominant transport, structure preservation, and severe mesh distortions make robust simulation difficult. We present a holistic finite element framework that jointly addresses these obstacles for biophysical applications by combining model-agnostic structure-preserving postprocessing, ALE-based mesh redistribution strategies driven by surface-tangential velocities, and stabilized discretization for advection-diffusion-reaction problems tailored to evolving domains. The methodology is modular and applies to advection-diffusion-reaction systems, Cahn-Hilliard phase separation, Helfrich-type geometric flows, as well as their staggered and potentially mixed-dimensional couplings. We provide a concise notation for evolving bulk and surface geometries, extend positivity-, bound-, and mass-preserving projections to moving meshes, and develop a two-step redistribution procedure that maintains element quality without remeshing. Convergence studies, manufactured solutions, and biologically motivated test cases -- including tumor-growth surrogates and phase segregation on deformable membranes -- demonstrate accuracy, stability, and versatility across the problem classes considered.