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A convergent evolving finite element algorithm for Willmore flow of closed surfaces

机译：一种闭合面Willmore流动的收敛演化有限元算法

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摘要

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.
机译：对闭合二维曲面的Willmore流和曲面扩散流的新型演化曲面有限元半离散化给出了收敛性的证明。这里提出和研究的数值方法将法向量和平均曲率的四阶演化方程离散化，重新表述为二阶方程组，并在插值到有限元空间的速度定律中使用这些演化几何量。该数值方法允许在多项式次数至少为两个的连续有限元的情况下进行收敛分析。误差分析结合了稳定性估计值和一致性估计值，为计算出的表面位置、速度、法向量和平均曲率生成了最优阶 H-1 范数误差边界。稳定性分析基于有限元法的矩阵向量公式，不使用几何参数。几何图形仅输入到一致性估计中。数值实验对理论结果进行了说明和补充。

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