Structure preserving discretisations of gradient flows for axisymmetric two-phase biomembranes

DOI10.1093/IMANUM/DRAA027zbMATH Open1516.65128arXiv2003.03127MaRDI QIDQ6336244

Publication date: 6 March 2020

Abstract: The form and evolution of multi-phase biomembranes is of fundamental importance in order to understand living systems. In order to describe these membranes, we consider a mathematical model based on a Canham--Helfrich--Evans two-phase elastic energy, which will lead to fourth order geometric evolution problems involving highly nonlinear boundary conditions. We develop a parametric finite element method in an axisymmetric setting. Using a variational approach, it is possible to derive weak formulations for the highly nonlinear boundary value problems such that energy decay laws, as well as conservation properties, hold for spatially discretised problems. We will prove these properties and show that the fully discretised schemes are well-posed. Finally, several numerical computations demonstrate that the numerical method can be used to compute complex, experimentally observed two-phase biomembranes.

Mathematics Subject Classification ID

Finite element methods applied to problems in solid mechanics (74S05) Variational methods applied to PDEs (35A15) PDEs in connection with biology, chemistry and other natural sciences (35Q92) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Finite difference methods for initial value and initial-boundary value problems involving PDEs (65M06) Iterative numerical methods for linear systems (65F10) Biomechanics (92C10) Membranes (74K15) Biomechanical solid mechanics (74L15) Computational methods for problems pertaining to biology (92-08) PDEs in connection with mechanics of deformable solids (35Q74) Preconditioners for iterative methods (65F08) Axially symmetric solutions to PDEs (35B07)

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