论文标题
平均曲率流量和封闭表面扩散的相互作用的数值分析
Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces
论文作者
论文摘要
分析了不断发展的表面有限元离散化，以通过耦合一般强制平均曲率流动和反应 - 扩散过程的系统控制的闭合二维表面的演变，这是受耦合能量的梯度流的启发。提出了两种算法，这两种算法都是基于一个系统，将扩散方程与表面速度定律中几何量的进化方程耦合。 事实证明，其中一种数值方法是在$ h^1 $ norm中收敛的，最佳订单对于有限元素至少两个。 我们提出了数字实验，说明了融合行为并证明了流动的定性特性：平均凸度，凸度丧失，弱最大原则和自我交流的发生。
An evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction--diffusion process on the surface, inspired by a gradient flow of a coupled energy. Two algorithms are proposed, both based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface. One of the numerical methods is proved to be convergent in the $H^1$ norm with optimal-order for finite elements of degree at least two. We present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.
