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

In this article we propose models and a numerical method for patternformation on evolving curved surfaces. We formulate reaction-diffusion equations onevolving surfaces using the material transport formula, surface gradients and diffusiveconservation laws. The evolution of the surface is defined by a material surface veloc-ity. The numerical method is based on the evolving surface finite element method. Thekey idea is based on the approximation of by a triangulated surface hconsistingof a union of triangles with vertices on . A finite element space of functions is thendefined by taking the continuous functions on hwhich are linear affine on each sim-plex of the polygonal surface. To demonstrate the capability, flexibility, versatility andgenerality of our methodology we present results for uniform isotropic growth as wellas anisotropic growth of the evolution surfaces and growth coupled to the solutionof the reaction-diffusion system. The surface finite element method provides a robustnumerical method for solving partial differential systems on continuously evolvingdomains and surfaces with numerous applications in developmental biology, tumourgrowth and cell movement and deformation.