Finite Element Method And Time Discretization For Allen-Cahn Equation

The Allen-Cahn equation is one of the most basic equations to describe the phase field model which is used to simulate the phase separation of binary alloys at a certain temperature.It has been widely applied to various problems such as crystal growth?phase transitions?image analysis?grain growth?and interfacial dynamics in material science.In addition,it is not easy to obtain the exact solution of the Allen-Cahn equation which is complex in practical problems.Therefore,it becomes particularly important how to solve the Allen-Cahn equation numerically.In Chapter 3,to solve the Allen-Cahn equation numerically,the semi-discrete scheme is obtained by using linear finite element in spatial discretization.An er-ror estimate between solution of the semi-discrete scheme and the exact solution is given by V.Thomee in the relevant monographs.Then a time-stepping scheme is constructed in time direction and a fully discrete scheme is obtained for the nu-merical solution of the Allen-Cahn equation.The unconditionally energy stability of the fully discrete scheme is proved and the sufficient condition for the existence of the unique solution for the fully discrete scheme is also given.Finally,an error estimate for the fully discrete scheme is established.The numerical experiments strongly prove our theoretical results:The fully discrete scheme is shown to be unconditionally energy stable and second order accurate both in time and space discretizations.In Chapter 4,in order to get the numerical solution of the Allen-Cahn equa-tion,the linear finite element is still used for the spatial discretization.Then stiff ordinary differential equations are obtained.Three different numerical schemes are utilized to solve this stiff ordinary differential equations.They are diagonal-ly implicit Runge-Kutta(DIRK)method?backward differential formulas(BDF)and backward differential formulas with scalar auxiliary variables(SAVBDF).The construction and stability of these three numerical schemes are discussed in detail.Finally,in the numerical experiments,the performances of these three different numerical schemes for solving the Allen-Cahn equation are compared.In Chapter 5,an adaptive finite element method for solving the Allen-Cahn equation and the Cahn-Hilliard equation is constructed.It is known to all that the Allen-Cahn equation and the Cahn-Hilliard equation can be regarded as singular perturbations of heat equation and biharmonic heat equation,respectively.It is natural to use adaptive grids instead of uniform grids to solve these two equations in order to improve computational efficiency.The numerical experiments show that when the zero level set of the solution moves with time,the refined grid also moves.The adaptive mesh graph shows that the recovery type a posterior error estimator can effectively guide the mesh to be adaptively refined near the interface.