Dissertation

2017

#### Degree Name

Doctor of Philosophy

Mathematics

#### Major

Applied Mathematics

Steven Hou

Huaiqing Wu

#### Abstract

The dissertation proposes and analyzes an efficient second-order in time numerical approximation for the Allen-Cahn equation, which is a nonlinear singular perturbation of the reaction-diffusion model arising from phase separation in alloys. We first present a fully discrete, nonlinear interior penalty discontinuous Galerkin (IPDG) finite element method, which is based on the modified Crank-Nicolson scheme and a mid-point approximation of the potential term $f(u)$. We then derive the stability analysis and error estimates for the proposed IPDG finite element method under some regularity assumptions on the initial function $u_0$. There are two key steps in our analysis: one is to establish an unconditionally energy-stable scheme for the discrete solutions; the other is to use a discrete spectrum estimate to handle the midpoint of the discrete solutions $u^m$ and $u^{m+1}$ in the nonlinear term, instead of using the standard Gronwall inequality technique. We obtain that all our error bounds depend on reciprocal of the perturbation parameter $\epsilon$ only in some lower polynomial order, instead of exponential order.

The dissertation also studies the stochastic Allen-Cahn equation, adding a white noise term to the right hand side of the deterministic Allen-Cahn equation. Three numerical experiments are performed with different initial conditions to study the evolution results of the stochastic case and compare these results with those of the deterministic case.

Junzhao Hu

en

application/pdf

83 pages

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