Numerical Methods for Deterministic and Stochastic Phase Field Models of Phase Transition and Related Geometric Flows

This dissertation consists of three integral parts with each part focusing on numerical approximations of several partial differential equations (PDEs). The goals of each part are to design, to analyze and to implement continuous or discontinuous Galerkin finite element methods for the underlying PDE problem. Part One studies discontinuous Galerkin (DG) approximations of two phase field models, namely, the Allen-Cahn and Cahn-Hilliard equations, and their related curvature-driven geometric problems, namely, the mean curvature flow and the Hele-Shaw flow. We derive two discrete spectrum estimates, which play an important role in proving the sharper error estimates which only depend on a negative power of the singular perturbation parameter ε [epsilon] instead of an exponential power. It is also proved that the zero level sets of the numerical solutions of the Allen-Cahn equation and the Cahn-Hilliard equation approximate the mean curvature flow and the Hele-Shaw flow respectively. Numerical experiments are carried out to verify the theoretical results and to compare the zero level sets of the numerical solutions and the geometric motions. Part Two focuses on finite element approximations of stochastic geometric PDEs including the phase field formulation of a stochastic mean curvature flow and the level set formulation of the stochastic mean curvature flow. Both formulations give PDEs with gradient-type multiplicative noises. We establish PDE energy laws and the Hölder [Holder] continuity in time for the exact solutions. Moreover, optimal error estimates are derived, and various numerical experiments are carried out to study the interplay of the geometric evolution and gradient-type noises. Part Three studies finite element methods for a quasi-static model of poroelasticity, which is a fluid-solid interaction multiphysics system at pore scale. We reformulate the original multiphysics system into a new system which explicitly reveals the diffusion process and has a built-in mechanism to overcome the locking phenomenon . Fully discrete finite element methods are proposed for approximating the new system. We derive a discrete energy law and optimal error estimates for our finite element methods. Numerical experiments are also provided to verify the theoretical results and to confirm that the locking phenomenon has indeed been overcome

Adaptive discontinuous Galerkin approximations to fourth order parabolic problems

An adaptive algorithm, based on residual type a posteriori indicators of errors measured in $L^{\infty}(L^2)$ and $L^2(L^2)$ norms, for a numerical scheme consisting of implicit Euler method in time and discontinuous Galerkin method in space for linear parabolic fourth order problems is presented. The a posteriori analysis is performed for convex domains in two and three space dimensions for local spatial polynomial degrees $r\ge 2$. The a posteriori estimates are then used within an adaptive algorithm, highlighting their relevance in practical computations, which results into substantial reduction of computational effort

C0 Interior Penalty Methods for Cahn-Hilliard Equations

In this work we study C0 interior penalty methods for Cahn-Hilliard equations. In Chapter 1 we introduce Cahn-Hilliard equations and the time discretization that leads to linear fourth order boundary value problems. In Chapter 2 we review related fundamentals of finite element methods and multigrid methods. In Chapter 3 we formulate the discrete problems for linear fourth order boundary value problems with the boundary conditions of the Cahn-Hilliard type, which are called C0 interior penalty methods, and we carry out the convergence analysis. In Chapter 4 we consider multigrid methods for the C0 interior penalty methods. We present two smoothing schemes and compare their performance. In Chapter 5 we apply the C0 interior penalty methods and the time discretization scheme to nonlinear time-dependent Cahn-Hilliard equations. Numerical examples for phase separation and image processing are presented