Abstract
In this paper, we devise a moving mesh finite element method for the approximate solution of coupled bulk–surface reaction–diffusion equations on an evolving two dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a novel moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known analytical solutions; these experiments indicate second-order spatial and temporal accuracy. Coupled bulk–surface problems occur frequently in many areas; in particular, in the modelling of eukaryotic cell migration and chemotaxis. We apply the method to a model of the two-way interaction of a migrating cell in a chemotactic field, where the bulk region corresponds to the extracellular region and the surface to the cell membrane.
Highlights
Coupled bulk–surface problems arise in many areas of engineering and the applied and natural sciences
In [40,41] we developed a “pseudopod-centered” [25] model based on a system of reaction–diffusion equations that gives rise to a suitable spatiotemporal activator profile that can be used for the generation of pseudopods without the need for a driving external signal
We have developed a computational framework for the solution of coupled bulk–surface reaction–diffusion equations in two dimensions
Summary
Coupled bulk–surface problems arise in many areas of engineering and the applied and natural sciences. It is well appreciated that migrating cells have the ability to shape external chemotactic fields through the use of membrane-bound enzymes that degrade the ligand field and by the self-secretion of chemoattractants that allow signal relay to neighbouring cells [50] The modelling of these important effects require the computational ability of solve partial differential equations on evolving bulk–surface domains. An advantage of the finite element framework is it allows the natural incorporation of flux boundary conditions linking the solution components in the bulk and surface domains Another potential advantage of the ALE approach is the ability to accommodate arbitrary mesh movements which are not necessarily Lagrangian.
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