Volume-preserving parametric finite element methods for axisymmetric geometric evolution equations

TL;DR

This paper introduces volume-preserving parametric finite element methods for axisymmetric surface evolution equations, ensuring volume conservation, stability, and efficiency in numerical simulations.

Contribution

It develops novel implicit finite element schemes that conserve volume and are unconditionally stable for axisymmetric geometric flows.

Findings

Methods are volume-preserving and unconditionally stable.

Numerical results demonstrate high accuracy and efficiency.

Schemes effectively handle surface diffusion and mean curvature flow.

Abstract

We propose and analyze volume-preserving parametric finite element methods for surface diffusion, conserved mean curvature flow and an intermediate evolution law in an axisymmetric setting. The weak formulations are presented in terms of the generating curves of the axisymmetric surfaces. The proposed numerical methods are based on piecewise linear parametric finite elements. The constructed fully practical schemes satisfy the conservation of the enclosed volume. In addition, we prove the unconditional stability and consider the distribution of vertices for the discretized schemes. The introduced methods are implicit and the resulting nonlinear systems of equations can be solved very efficiently and accurately via the Newton's iterative method. Numerical results are presented to show the accuracy and efficiency of the introduced schemes for computing the considered axisymmetric geometric flows.