A Structure-Preserving Parametric Finite Element Method for Surface Diffusion

We propose a structure-preserving parametric finite element method (SP-PFEM) for discretizing the surface diffusion of closed curve in two dimensions (2D) or three (3D). Here "structure-preserving" refers to preserving fundamental geometric structures flow: (i) conservation area/volume enclosed by curve/surface, and (ii) decrease perimeter/total area curve/surface. For simplicity notations, we begin with 2D present weak (variational) formulation governing equation. Then discretize variational using backward Euler time piecewise linear elements space, proper approximation unit normal vector information curves at current next step. The constructed numerical is shown preserve also enjoys good property asymptotic equal mesh distribution. proposed SP-PFEM "weakly" implicit (or almost semi-implicit) nonlinear system each step can be solved very efficiently accurately Newton's iterative method. then extended 3D. Extensive results, including convergence tests, distribution, are reported demonstrate accuracy efficiency simulating