论文标题
封闭表面的广义平均曲率流的收敛有限元算法
A convergent finite element algorithm for generalized mean curvature flows of closed surfaces
论文作者
论文摘要
提出了一种算法，用于封闭的二维表面的广义平均曲率流，其中包括平均曲率流，平均值和平均平均曲率流量等。证明了误差估计值，用于全面流动的半离散和完全离散。在此提出和研究的算法结合了不断发展的表面有限元，其节点决定了离散表面，并线性隐式向后差进行时间整合。数值方法基于一个系统，该系统将表面演变与正常速度和正常矢量的非线性二阶抛物线进化方程相连。在多项式程度有限元的情况下，提出了融合证明，至少有两个至5的订单差异公式。误差分析结合了稳定性估计和一致性估计，以产生最佳阶$ h^1 $ norm误差界，用于计算的表面位置，速度，正常矢量，正常速度，因此对于平均曲率。稳定性分析是在矩阵矢量公式中进行的，并且与仅输入一致性分析的几何参数无关。提出了数值实验，以说明收敛结果，并报告单调量，例如〜霍金质量的反向平均曲率流。补充非凸表面的实验。
An algorithm is proposed for generalized mean curvature flow of closed two-dimensional surfaces, which include inverse mean curvature flow, powers of mean and inverse mean curvature flow, etc. Error estimates are proven for semi- and full discretisations for the generalized flow. The algorithm proposed and studied here combines evolving surface finite elements, whose nodes determine the discrete surface, and linearly implicit backward difference formulae for time integration. The numerical method is based on a system coupling the surface evolution to non-linear second-order parabolic evolution equations for the normal velocity and normal vector. Convergence proofs are presented in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. The error analysis combines stability estimates and consistency estimates to yield optimal-order $H^1$-norm error bounds for the computed surface position, velocity, normal vector, normal velocity, and therefore for the mean curvature. The stability analysis is performed in the matrix-vector formulation, and is independent of geometric arguments, which only enter the consistency analysis. Numerical experiments are presented to illustrate the convergence results, and also to report on monotone quantities, e.g.~Hawking mass for inverse mean curvature flow. Complemented by experiments for non-convex surfaces.
