Abstract:In this paper, we propose a second-order-in-time, structure-preserving, and mesh-robust parametric finite element method for surface diffusion and volume-preserving mean curvature flow. We first reformulate the original evolution equations into new systems in which the tangential motion is governed by a harmonic map heat flow. This heat flow maps a fixed reference surface onto the unknown evolving surface and drives points on the evolving surface to move in their tangent spaces so as to reduce the associated harmonic energy. As a result, in the discrete setting, the mesh quality can be maintained at a level comparable to that of the reference surface, unless singularities occur. The volume-preserving property is theoretically guaranteed by the careful design of the scheme, while energy dissipation is enforced through a Lagrange multiplier. We present several numerical experiments to demonstrate second-order convergence in time and the advantage of the proposed method in preserving mesh quality. The structure-preserving properties are further confirmed by the numerical results. Finally, the proposed framework can be readily extended to other geometric flows.
From: Zongze Yang [view email]
[v1]
Sat, 6 Jun 2026 18:29:15 UTC (5,865 KB)
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