A finite element method for surface diffusion: the parametric case
- 作者：B脙脙nsch ; Eberhard ; Morin ; Pedro ; Nochetto ; Ricardo H.
- 关键词：35K55 ; 65M12 ; 65M15 ; 65M60 ; 65Z05 ; Surface diffusion ; Fourth-order parabolic problem ; Finite elements ; Schur complement ; Smoothing effect ; Pinch-off
- 刊名：Journal of Computational Physics
- 出版年：2005
- 期刊代码：136_00219991
- 类别：cp
- 出版时间：February 10, 2005
- 卷：203
- 期：1
- 页码：321-343
- 文件大小：3.53 M
摘要
Surface diffusion is a (fourth order highly nonlinear) geometric driven motion of a surface with normal velocity proportional to the surface Laplacian of mean curvature. We present a novel variational formulation for parametric surfaces with or without boundaries. The method is semi-implicit, requires no explicit parametrization, and yields a linear system of elliptic PDE to solve at each time step. We next develop a finite element method, propose a Schur complement approach to solve the resulting linear systems, and show several significant simulations, some with pinch-off in finite time. We introduce a mesh regularization algorithm, which helps prevent mesh distortion, and discuss the use of time and space adaptivity to increase accuracy while reducing complexity.