Weak Galerkin Finite Element Methods For Some Phase Field Models With Nonuniform Second-Order Schemes

Weak Galerkin finite element method is an effective and robust numerical method for partial differential equations.The core and essence of it are to intro-duce the so-called weak functions and weak operators.By choosing appropriate function spaces for the weak functions and weak operators,the weak Galerkin finite element method provides credible approximations for partial differential equaitons.Phase field models,also known as the interface diffusion model,has many significance applications in solidification kinetics,crack growth simulation and etc.It is well-known that the phase field models involve different time scales and possess a property that they admit the energy dissipation law.This thesis is concerned with the application of the weak Galerkin finite ele-ment method to the phase field models with a nonuniform BDF2 scheme and a L1+ formula.The numerical schemes we established in this thesis all satisfy dis-crete energy stability and nonuniform time-step is used in the numerical schemes to reduce the computational cost,adpative time-step strageties were used in nu-merical experiments.In Chapter 1,we introduce some properties and present a literature review of the weak Galerkin finite element method.Basic properties of the phase field models and corresponding numerical methods are introduced too.In Chapter 2,we present some basic notations and introduced the model equa-tion,the BDF2 formula,the L1+ formula and two different adptive time-step strageties.In Chapter 3,on spatial discretization we use the lowest-order weak Galerkin finite element method and on temporal discretization we adopt the nonuniform BDF2 formula to solve the Allen-Calm equation.Related energy stability analysis and optimal orcler error estimates of L2 norm are establishe·.Several numerical experiments are illustrated and adaptive time-step strageties are used.In Chapter 4,on spatial discretization we use the lowest-order weak Galerkin finite element method and adopt the nonuniform L1+ formula to discretize the Caputo fractional derivative in order to solve the time-fractional Allen-Cahn equa-tion.Corresponding discrete energy stability analysis are established.Several numerical experiments are illustrated,fast evaluation scheme and adaptive time-step strageties are used.In Chapter 5,on spatial discretization we use the stabilizer-free weak Galerkin finite element method and on temporal discretization we employ the nonuniform BDF2 formula to solve the Cahn-Hilliard equation.Related energy stability anal-ysis are established.Several numerical experiments are illustrated and adaptive time-step strageties are used.In Chapter 6,we present some conclusions of this thesis and some prospects of future work.