A Narrow Band Numerical Method for a Surface Reaction-Diffusion System Coupled with Surface Motion

Reaction-diffusion equations on surfaces are widely used for modeling various phenomena in biology. This paper presents a novel numerical method for solving a surface reaction-diffusion system coupled with the evolution of the surface. The coupled system has been used to model the growth of hard tumors. A stabilized trace finite element method is used to discretize the reaction-diffusion system on evolving surfaces. The surface motion is computed using a diffusion-generated method for the level-set function, which involves solving a heat equation in each time step followed by a redistance operation. Both the trace finite element space for the reaction-diffusion system and the finite element space for the level-set function are defined in a narrow band region near the surface on a bulk mesh. The method is fully decoupled and allows for easy handling of topology changes. Numerical experiments demonstrate the efficiency of the proposed method for solving this complex problem.