Structure-preserving Lagrange-multiplier methods for mean curvature flow and their error bounds
This work provides a structure-preserving numerical method for mean curvature flow, which is important for applications in geometry processing and materials science, but the approach is incremental as it builds on existing evolving surface finite element methods.
The paper proposes a class of evolving surface finite element methods for mean curvature flow that preserve the energy-decreasing structure using Lagrange multipliers. The methods achieve optimal-order error bounds of order $h^k + \tau^q$ in the $H^1$-norm and are area-decreasing at every time step, with numerical experiments confirming predicted convergence rates.
We propose and analyze a class of evolving surface finite element methods for mean curvature flow of closed surfaces using Lagrange multipliers for preserving the energy-decreasing structure. The algorithm is based on the solution-driven formulation using Huisken's evolution equations for the normal vector and the mean curvature. The time discretizations use linearly implicit backward difference formulas (BDF) for the parabolic geometric variables and implicit Adams updates for the nodal positions. The approach also accommodates artificial tangential velocities of minimal-deformation-rate type. The resulting fully discrete algorithms are area-decreasing at every time step, with a prescribed decay rate determined by the computed mean curvature. We prove local existence and uniqueness of the discrete Lagrange multiplier and convergence of a simplified Newton iteration for its computation under weak regularity assumptions. Under stronger regularity assumptions, as used in the convergence theory for the underlying evolving surface finite element method, we derive optimal-order error bounds of order $h^k+τ^q$ in the $H^1$-norm for finite elements of polynomial degree $k\ge 2$ and $q$-step BDF and $q$-step implicit Adams methods with $2\le q\le 5$, both without and with minimal-deformation-rate tangential motion. Numerical experiments for mean curvature flow of a sphere confirm the predicted convergence rates and show that the Lagrange-multiplier correction entails only a small computational overhead that is essentially independent of the mesh size and the time step size.