Error Analysis of Finite Element Approximation for Mean Curvature Flows in Axisymmetric Geometry

Abstract

We focus on establishing a novel convergence analysis of the numerical approximation methods applied to axisymmetric mean curvature flows (MCFs) with genus-1 surfaces, including both isotropic and anisotropic cases. This advancement builds upon and enhances the findings presented in a previous work in Barrett et al. (IMA J Numer Anal 41:1641–1667, 2021), which exists a strict restriction on the time-space step ratio, i.e. $\Delta t \lesssim \sqrt{h}$. In this work, we first adopt a novel time-space error splitting technique to remove above restriction under the isotropic case. Furthermore, we discover that this splitting also aids in addressing another unresolved issue present in anisotropic MCFs. Indeed, almost any restrictions on the time and space steps cannot derive the $L^\infty$-boundedness of $D_t\vec {X}_{h, \rho }^m$, which is required for convergence analysis in the anisotropic case. Given that $\Vert D_t\vec {X}_{h, \rho }^m\Vert _{L^\infty } \lesssim \Delta t^{-1}h^{-1/2}(\Delta t + h) + 1$, we need to ensure that $h^{-1/2}(1 + \Delta t^{-1}h) \lesssim 1$. This condition is not feasible unless $h > 1$ is required. Within the scope of this research, we demonstrate that the error analytical technique established for the isotropic case, is also helpful for the anisotropic cases. Finally, we showcase a variety of numerical experiments, including the convergence tests to validate the theoretical analysis, as well as numerical simulations to to further support our findings.