Graduate Student Organization Seminar

Date: January 29, 2020

Time: 4:00PM - 5:00PM

Location: BLOC 628

Speaker: Jiuhua Hu

Title: Homogenization of time-fractional diffusion equations with periodic coefficients

Abstract: In this talk, I will begin with introducing one of the model reduction techniques: homogenization. We shall consider the initial boundary value problem for the time-fractional diffusion equation with a homogeneous Dirichlet boundary condition and an inhomogeneous initial data $a(x)\in L^{2}(D)$ in a bounded domain $D\subset \mathbb{R}^d$ with a sufficiently smooth boundary. We analyze the homogenized solution under the assumption that the diffusion coefficient $\kappa^{\epsilon}(x)$ is smooth and periodic with the period $\epsilon>0$ being sufficiently small. We derive that its first order approximation measured by both pointwise-in-time in $L^2(D)$ and $L^p((\theta,T); H^1(D))$ for $p\in [1,\infty)$ and $\theta\in (0,T)$ has a convergence rate of $\mathcal{O}(\epsilon^{1/2})$ when the dimension $d\leq 2$ and $\mathcal{O}(\epsilon^{1/6})$ when $d=3$. Several numerical tests are presented to demonstrate the performance of the first order approximation. This is joint work with Guanglian Li.