A convergent algorithm for forced mean curvature flow driven by diffusion on the surface

Abstract

The evolution of a closed two-dimensional surface driven by both mean curvature flow and a reaction–diffusion process on the surface is formulated as a system that couples the velocity law not only to the surface partial differential equation but also to the evolution equations for the normal vector and the mean curvature on the surface. Two algorithms are considered for the obtained system. Both methods combine surface finite elements for space discretization and linearly implicit backward difference formulae for time integration. Based on our recent results for mean curvature flow, one of the algorithms directly admits a convergence proof for its full discretization in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five, with optimal-order error bounds. Numerical examples are provided to support and complement the theoretical convergence results (illustrating the convergence behaviour of both algorithms) and demonstrate the effectiveness of the methods in simulating a three-dimensional tumour growth model.