Dynamic Ritz projection of mean curvature flow and optimal ${\mathbf{L}}^2$ convergence of parametric FEM

Abstract

A new approach is developed to study the convergence of parametric finite element approximations to the mean curvature flow of closed surfaces in three-dimensional space. In this approach, the error analysis is conducted by comparing the numerical solution to a dynamic Ritz projection of the mean curvature flow introduced in this paper, rather than an interpolation of the mean curvature flow, as commonly used in the literature. The errors associated with the dynamic Ritz projection in approximating the mean curvature flow are established in the $L^2$ and $W^{1,p}$ norms. Leveraging these results, optimal-order convergence of parametric finite element methods for mean curvature flow of closed surfaces in the $L^{\infty }(0,T;L^2)$ norm is proved, including the convergence of parametric finite element methods with piecewise linear finite elements.