| Abstract: |
| We propose a novel formulation for parametric finite element methods to simulate
surface diffusion of closed curves, which is also called as the curve diffusion. Several high-order
temporal discretizations are proposed based on this new formulation. To ensure that the numerical methods preserve geometric structures of curve diffusion (i.e., the perimeter-decreasing and area-preserving properties), our formulation incorporates two scalar Lagrange multipliers and two evolution equations involving the perimeter and area, respectively. By discretizing the spatial variable using piecewise linear finite elements and the temporal variable using either the Crank-Nicolson method or the backward differentiation formulae method, we develop high-order temporal schemes that efectively preserve the structure at a fully discrete level. These new schemes are implicit and can be efficiently solved using Newtons method. Extensive numerical experiments demonstrate that
our methods achieve the desired temporal accuracy, as measured by the manifold distance, while
simultaneously preserving the geometric structure of the curve diffusion. |
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