In this talk, we will introduce our recent advances in the multiscale analysis and algorithms for the electromagnetic properties of novel materials and devices. On Part one, we will discuss the algorithm for the Boltzmann transport equation (BTE) with multi-valley arising from the modeling of the carriers transport in semiconductor materials and devices. Some theoretical results such as the consistency between the solution of the BTE and a Markov process, the invariant measure and well-posedness for the solution of the BTE are proved. A DSMC algorithm and the convergence results are proposed. The typical applications for some semiconductor materials such as GaAs and graphene are advanced. On Part two, we will present the homogenization method and the multiscale asymptotic method for the modified Maxwell-Dirac system in matrix form with rapidly oscillating discontinuous coefficients in a bounded Lipschitz convex domain under the Weyl gauge. The Maxwell-Dirac system with quadratic correction has many applications in materials science such as topological insulators, graphene, superconductors and so forth. Based on the multiscale asymptotic expansions of the solution of the Maxwell-Dirac system, by using the Crank-Nicolson finite difference method and the adaptive edge element method, we develop the multiscale algorithms for solving the Maxwell-Dirac system with rapidly oscillating discontinuous coefficients. Numerical results are then carried out to validate the proposed method.
This talk is based on the joint work with Jiachuan Cao and Yaoyao Fu

曹礼群，男，中国科学院数学与系统科学研究院研究员，博士生导师，现任中科院数学院计算数学与科学工程计算研究所副所长，中国计算物理学会常务理事，中国计算数学学会理事，中国计算力学专业委员会委员，北京市计算数学学会副理事长。研究方向：多尺度建模与算法。