论文标题
封闭表面的Willmore流动的收敛性有限元算法
A convergent evolving finite element algorithm for Willmore flow of closed surfaces
论文作者
论文摘要
为新颖的封闭二维表面的Willmore流以及表面扩散流的新型表面有限元的半差异提供了收敛的证明。此处提出和研究的数值方法将正常矢量和平均曲率的四阶演化方程离散，并将其重新构成二阶方程系统，并使用这些速度定律中的这些不断发展的几何量，这些几何量被插入有限元元素空间。在多项式程度的连续有限要素的情况下，这种数值方法接受了收敛分析。误差分析结合了稳定性估计和一致性估计值，以产生最佳订购$ h^1 $ -Norm误差界，用于计算的表面位置，速度，正常向量和平均曲率。稳定性分析基于有限元方法的矩阵 - 向量公式，并且不使用几何参数。几何形状仅进入一致性估计。数值实验说明并补充理论结果。
A proof of convergence is given for a novel evolving surface finite element semi-discretization of Willmore flow of closed two-dimensional surfaces, and also of surface diffusion flow. The numerical method proposed and studied here discretizes fourth-order evolution equations for the normal vector and mean curvature, reformulated as a system of second-order equations, and uses these evolving geometric quantities in the velocity law interpolated to the finite element space. This numerical method admits a convergence analysis in the case of continuous finite elements of polynomial degree at least two. 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 and mean curvature. The stability analysis is based on the matrix--vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments illustrate and complement the theoretical results.
