Table of Contents:
1 The Curve Shortening Flow 2
1.1 Some Definitions and Notations Concerning Plane Closed Curves 2
1.2 Definition of the Curve Shortening Process 3
1.3 Self-Similar Solutions of the Curve Shortening Process 6
2 Mean Curvature Flow for Hypersurfaces in $R^n$ 7
2.1 Some Definitions and Notations for Hypersurfaces 7
2.2 Formulation of the Mean Curvature Flow Involving the Laplace--Beltrami Operator 9
2.3 The Level Set Formulation of Mean Curvature Flow and the Formulation for Graphs 10
2.4 Viscosity Solutions for the Level Set Formulation 11
2.4.1 Generalized Motion by Mean Curvature 11
2.4.2 Some Properties of Weak Solutions 12
2.4.3 Existence of Weak Solutions 12
2.5 MCF via the Allen-Cahn Equation 14
2.5.1 The Distance Function 14
2.5.2 The Allen--Cahn Equation 15
2.5.3 Asymptotic Analysis for the Allen-Cahn Equation 16
3 Numerical Methods for Mean Curvature Flow 17
3.1 Numerical Treatment of the Curve Shortening Flow 17
3.2 A Numerical Method for Mean Curvature Flow Using the Laplace--Beltrami Operator 19
3.3 Numerical Methods for MCF Based on the Level Set Formulation 20
3.3.1 Hyperbolic Method 20
3.3.2 Parabolic Method 20
3.4 A Numerical Method for MCF Using the Allen--Cahn Equation 21
3.5 Asymptotic Convergence for a Finite Element Method for the MCF of Graphs 21
4 The Numerical Computation of Threedimensional Dendrites 23
4.1 The physical model 24
4.2 Formulation of the numerical method 26
4.2.1 The free boundary $\Gamma (t)$ 27
4.2.1.1 Time discretization 27
4.2.1.2 Weak formulation 28
4.2.1.3 Triangulation of the manifold and finite elements 28
4.2.1.4 Finite element formulation 29
4.2.2 Heat equation 29
4.2.2.1 Time discretization 30
4.2.2.2 Finite elements 30
4.3 Numerical methods 31
4.3.1 Heat equation 31
4.3.2 Numerical methods for the free boundary 32
4.3.2.1 Deformation of the triangulation 32
4.3.2.2 Error estimator 34
4.3.2.3 Local refinement and coarsening 35
4.3.2.4 Node displacement on the surface 35
4.3.3 Adaptive algorithm for the crystal growth problem 35
4.4 Numerical results 36
5 References 40