The surface finite element method for pattern formation on evolving biological surfaces

Abstract

In this article we propose models and a numerical method for pattern formation on evolving curved surfaces. We formulate reaction-diffusion equations on evolving surfaces using the material transport formula, surface gradients and diffusive conservation laws. The evolution of the surface is defined by a material surface velocity. The numerical method is based on the evolving surface finite element method. The key idea is based on the approximation of Γ by a triangulated surface Γ(h) consisting of a union of triangles with vertices on Γ. A finite element space of functions is then defined by taking the continuous functions on Γ(h) which are linear affine on each simplex of the polygonal surface. To demonstrate the capability, flexibility, versatility and generality of our methodology we present results for uniform isotropic growth as well as anisotropic growth of the evolution surfaces and growth coupled to the solution of the reaction-diffusion system. The surface finite element method provides a robust numerical method for solving partial differential systems on continuously evolving domains and surfaces with numerous applications in developmental biology, tumour growth and cell movement and deformation.

Numerical Preservation of Velocity Induced Invariant Regions for Reaction–Diffusion Systems on Evolving Surfaces

“…, N . The LESFEM method (41) differs from the evolving surface finite element method (ESFEM) in [1] due to the presence of the interpolant operator on the first and the last terms in (41). By expressing U 1 , .…”

“…Following the evolving surface finite element method (ESFEM) studied in [1] for the approximation of the variational problem (29), we present its lumped counterpart, the lumped evolving surface finite element method (LESFEM).…”

“…, r , be their fluxes tangent to Γ (t). We recall from [1] the derivation of a system of r equations for u := (u 1 , . .…”

“…Following [1], we derive the variational formulation of system (16). To this end, for each t ∈ [0, T ] we multiply equations (16) by the respective test functions…”

“…The velocity field (48) corresponds to the specific case of isotropic evolution, see for instance [8,29] for the case of evolving planar domains and [1,32] for the general case. In particular, for suitable choices of the function S(t), the growth law (48) admits some specific cases such as uniform, exponential, logistic and periodic growth, see [1,8,29,32]. From (3), it is easy to show that, with the velocity field (48), each x 0 ∈ Γ 0 evolves to the point…”