间断Galerkin方法及在电磁计算中的应用研究

Discontinuous Galerkin (DG) methods can be viewed as a clever combination of the finite element methods (FEM) and the finite volume methods (FVM). Ideally, the DG methods share almost all the advantages of the FEM and the FVM. Recently, hybridizable discontinuous Galerkin (HDG) methods have been introduced. Compared to the DG methods, the HDG methods have less globally coupled degrees of freedom. Thus, the HDG methods are considered to be computationally cheaper than the DG methods. This project will employ the newly introduced HDG methods to solve freqency-domain electromagnetic problems. The purpose of this project is to make the numerical modelling for the electromagnetic wave propagation in complex and heterogeneous media more accurate and faster. We shall consider the optimized Schwarz algorithm under the framework of the HDG methods as well as a class of locally well-posed HDG methods. There are two practical ways for the implementation of the optimized Schwarz methods. The First one is directly followed from the optimized Schwarz algorithm with DG discretization. Only the information of the electric field and magnetic field will be exchanged on the interfaces between subdomains in this case. The other one takes the continuity of the hybrid term into consideration besides the electric field and the magnetic field. The local well-posedness of the HDG methods can be achieved by introducing a stabilization term living on the skeleton of the mesh. We will finally propose a stable and high-efficient methodology for the solution of the frequency-domain electromagnetic problems. And we will also produce a software package which can be widely used in electromagnetic modelling.

间断Galerkin(DG)方法在某种程度上可看成是有限体积法与有限元法的一种有机结合，兼具二者的优点。最近提出的杂交间断Galerkin(HDG)方法继承了DG方法的优点，且具有更少的全局自由度，因此期待HDG方法比经典DG方法效率更高。本项目将新型的HDG方法应用于频域电磁计算，进行相关理论和算法创新，以提高电磁模拟的精度和速度。主要研究内容包括：结合HDG离散的最优Schwarz算法；局部适定的HDG方法构造及其性能研究。实现最优Schwarz算法有两种途径：一是完全继承DG方法离散时的做法，不同之处只在于子区域内用HDG方法求解；二是将子区域间杂交项的连续性也考虑在内，即在子区域间交换电场和磁场相关信息的同时，也交换杂交项的信息。通过引入稳定项得到一类广泛的具有局部适定性与最优收敛性质的HDG方法。项目最终将形成一套求解复杂电磁环境下频域麦克斯韦方程的稳定、高效方法及软件包。

项目主要研究了求解时谐Maxwell方程的HDG方法，构造了一类广义的局部适定的HDG方法，给出了这类方法与“经典”DG与HDG方法的联系；设计了求解大规模电磁散射问题的优化Schwarz算法，指出了Schwarz类区域分解算法的实现只需调整HDG方法中的稳定性因子即可；同时也构造了求解HDG离散产生的线性代数系统的Krylov子空间方法和不完全分解预条件子；项目形成了相应的MATLAB与Fortran语言编写的软件包。已在国际权威期刊上发表SCI论文15篇。