Local Discontinuous Galerkin Methods And Fast Solvers For Phase Field Models

In this thesis, we develop local discontinuous Galerkin (LDG) methods for phase field models. The LDG method can achieve high order of accuracy and thus capture the generally sharp interfacial structures in phase field models. These equations consist of Cahn-Hilliard equation, Allen-Cahn equation, Cahn-Hilliard-Hele-Shaw system, Cahn-Hilliard-Brinkman system and functionalized Cahn-Hilliard. Phase field models are partial differential equations (PDEs) with high order space derivatives, which needs use implicit time discretization methods to remove severe time step restriction for explicit methods. To enhance the efficiency of the proposed approach, we use the multigrid method to solve algebraic systems. The energy stability of LDG methods is proved. The unconditional energy stability for convex splitting methods is also proved, which allows us choose time steps adaptively. In addition, we prove an a priori error estimates in L2norm and negative norm of the semi-discrete LDG method for the Allen-Cahn equation with smooth solution. This thesis is mainly divided into four parts.In the first part, we develop and analyze a fast solver for the system of algebraic equations arising from the LDG discretization and implicit time marching methods to the Cahn-Hilliard type equations with constant and degenerate mobility. The Cahn-Hilliard equation with degenerate mobility makes implicit time discretization methods are difficult to design and implementation. We introduce linearization technique and obtain a high order implicit time discretization method. The discrete energy stability for the Cahn-Hilliard equations with a special homogeneous free energy density (?)(u)(1/4)(1—u2)2is proved based on the convex splitting method. We can get the steady-state solution numerically with implicit time discretization methods and multigrid solver.In the second part, we present an LDG method for the Allen-Cahn equation and prove the energy stability. In addition, we prove an a priori error estimate in L2norm. The nonlinear term in Allen-Cahn equation makes error estimate more difficult. By special treatment for nonlinear term, we finally get the optimal convergence rate of k+1in the L2norm for k-th order polynomial approximation. We also achieve the convergence rate of2k+1in negative norm by employing a dual argument.In the third part, we develop LDG methods for fourth order nonlinear Cahn-Hilliard-Hele-Shaw system, Cahn-Hilliard-Brinkman system and sixth order nonlinear functionalized Cahn-Hilliard equation, respectively. The energy stability is proved. The high order nonlinear properties for these three equations pose a great deal of difficul-ty for numerical simulation. In addition, we also need to solve the coupled equation (?)·u=0in Cahn-Hilliard-Hele-Shaw system and Cahn-Hilliard-Brinkman system. Meanwhile, the severe time step restriction for explicit methods of these three equa-tions turns explicit methods useless. We introduce semi-implicit schemes which consist of the implicit Euler method combined with a convex splitting of the discrete Cahn-Hilliard energy strategy for the temporal discretization for Cahn-Hilliard-Hele-Shaw system and Cahn-Hilliard-Brinkman system, respectively. The unconditional energy stability for these fully-discrete LDG schemes are proved, respectively. It is a chal-lenge to solve the system of equations arised by space-time discretizations for these three equations.In this thesis, what we have considered are PDEs with even order spatial deriva-tives. In the fourth part, we develop a fast iterative solver for the system of linear equations arising from the LDG spatial discretization and additive Runge-Kutta (ARK) time marching method for the KdV type equations. The equations at the implicit time level are strongly non-symmetric, thus it is difficult to solve the equations. We rewrite the KdV type equations as a system of PDEs to relax the non-symmetric property. We apply the multigrid method to solve the equations and the local Fourier analysis is adopted to analyze the convergence of the multigrid solver.The multigrid method is used to solve the equations arised by LDG spatial dis-cretization and implicit time integration methods and we show numerically that the multigrid solver has optimal or nearly optimal convergence complexity, which is more efficient than traditional iterative methods. Numerical results for one-dimensional, two-dimensional and three-dimensional cases are given to illustrate the efficiency of the methods. In addition, we can get k+1-th order accuracy in both L2and L∞norms for k-th order polynomial approximation numerically.