高波数时谐Maxwell方程的高效数值解法

Computational electromagnetic is of considerable importance in many areas of engineering and science such as aerospaceindustry, microwave and millimeter-wave communication, radar and geological exploration.The efficient solution of the time-harmonic Maxwell equation has become an active research area in computational electromagnetic. For the problems with high wave numbers,numerical dispersion errors arise in the approximation of short unresolved waves, which induce the so-called pollution error. For the standard numerical methods, the pollution term has a major influence with standard mesh size in engineering. Moreover, due to the strong indefinitenessof the discrete algebra system, standard iterative algorithms will fail.The first objective of this research project is to design the stabilized discrete approximations for the time-harmonic Maxwell equation with high wave number in order to reduce the pollution error. Especially, we will focus on the hybridizable discontinuous Galerkin method and continuous interior penalty finite element method. We will also aim to propose the robust multilevel methods for the time-harmonic Maxwell equation with high wave numbers. We will utilize the stabilized discrete approximation to design efficient correction problems on the coarse grids and different mixed-type smoothers will be applied according to the mesh size. For the proposed HDG and CIP methods,we will further analyze the dependence of the convergence rateon the wave number, the mesh size and the polynomial degree.Besides, we will also design the corresponding robust multilevel methods for the discrete algebraic systems.

计算电磁学在工程科学中有着非常重要的作用，并且已被广泛应用于航空航天工业、微波与毫米波通信、雷达和地质勘探等领域中。时谐Maxwell方程的求解一直是计算电磁学中很活跃的研究分支。对于高波数问题，数值耗散会引起所谓的污染误差，工程计算中如果采用传统方法求解，污染误差将占据主要影响；其次，由于离散代数系统的强不定性，通常的迭代算法不再有效，收敛速度很慢或发散。本项目的第一个目标是针对高波数时谐Maxwell方程设计出稳定的数值离散格式以减少污染误差的影响，着重考虑杂交间断Galerkin有限元和连续内罚有限元；本项目还将研究求解高波数时谐Maxwell方程稳定的多水平方法，利用稳定的离散格式设计有效的粗空间校正问题，根据网格尺寸选择不同的混合型光滑子；在前述时谐Maxwell方程研究的基础上，分析高阶有限元的收敛速度与波数、网格尺寸以及多项式次数的关系，设计出相应离散代数系统的多水平算法。

计算电磁学在工程科学中有着非常重要的作用,并且已被广泛应用于航空航天工业、微波与毫 米波通信、雷达和地质勘探等领域中。时谐Maxwell方程的求解一直是计算电磁学中很活跃的 研究分支。对于高波数问题,数值耗散会引起所谓的污染误差,工程计算中如果采用传统方法 求解,污染误差将占据主要影响;其次,由于离散代数系统的强不定性,通常的迭代算法不再 有效,收敛速度很慢或发散。本项目的第一个目标是针对高波数时谐Maxwell方程设计出稳定 的数值离散格式以减少污染误差的影响,着重考虑杂交间断Galerkin有限元和连续内罚有限元 ;本项目还将研究求解高波数时谐Maxwell方程稳定的多水平方法,利用稳定的离散格式设计 有效的粗空间校正问题,根据网格尺寸选择不同的混合型光滑子;在前述时谐Maxwell方程研 究的基础上,分析高阶有限元的收敛速度与波数、网格尺寸以及多项式次数的关系,设计出相 应离散代数系统的多水平算法。