The evolution of a closed two-dimensional surface driven by both mean curvature flow and a reaction--diffusion process on the surface is formulated into a system, which couples the velocity law not only to the surface partial differential equation but also to the evolution equations for the geometric quantities, namely the normal vector and the mean curvature on the surface. Two algorithms are considered for the obtained system. Both methods combine surface finite elements as a space discretisation 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 discretisation in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. Numerical examples are provided to support and complement the theoretical convergence results (demonstrating the convergence properties of the method without error estimate), and demonstrate the effectiveness of the methods in simulating a three-dimensional tumour growth model.

中文翻译：

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表面扩散驱动的强迫平均曲率流收敛算法

由平均曲率流和表面上的反应-扩散过程共同驱动的封闭二维表面的演化被公式化为一个系统，该系统不仅将速度定律耦合到表面偏微分方程，而且还耦合到演化方程对于几何量，即表面上的法向量和平均曲率。为获得的系统考虑了两种算法。这两种方法都结合了表面有限元作为空间离散化和线性隐式后向差分公式进行时间积分。根据我们最近对平均曲率流的结果，在多项式次数至少为 2 的有限元和 2 到 5 阶的后向差分公式的情况下，其中一种算法直接承认其完全离散化的收敛证明。