高波数波动问题的区域分解法

Computational acoustics and electromagnetics are of considerable importance in the area of engineering and science, as a result, how to solve the Helmholtz equation and the time-harmonic Maxwell equation accurately and efficiently has become a popular research topic, especially for the high wave number case. Standard fast solvers have encountered convergence problems due to the strong indefiniteness of the resulting linear system for the discretizations of the Helmholtz equation and the time-harmonic Maxwell equation. Besides,the large nullspace of the curl-operator in Maxwell equation also gives rise to serious difficulties.In this project, we shall use the Robin-type domain decomposition method to efficiently slove the the Helmholtz equation and the time-harmonic Maxwell equation. Meanwhile, some stable finite elements methods like the continuous interior penalty finite element method (CIP) and discontinuous Petrov-Galerkin methods (DPG) shall be considered to design suitable coarse spaces in our domain decomposition methods. Numerical results supported our theoretical findings shall be given.

在科学工程计算领域, 计算声学和电磁学越来越重要, 如何高效求解它们的数学模型, 即Helmholtz方程和时谐Maxwell方程, 是近年来科学工程计算界非常关注的问题, 尤其是高波数情形。由于离散代数系统的强不定性, 标准的快速求解器将遭遇收敛性变慢或不收敛的问题,此外Maxwell方程中curl 算子巨大的核空间也给算法的设计带来了困难。本项目将利用Robin 型区域分解法来高效并行求解Helmholtz方程和时谐Maxwell方程, 此外一些稳定的有限元离散方法,例连续内点惩罚有限元方法(CIP)和间断Petrov-Galerkin 方法(DPG), 将用来构造有效的粗空间。大量的数值实验将验证我们算法的有效性。

本项目主要研究高波数Helmholtz方程和时谐Maxwell方程的数值求解，此类问题在科学和工程计算邻域非常重要并具有很大的挑战性。由于离散代数系统的强不定性, 标准的求解器，例如Jacobi, Gauss-Seidal 和 CG方法，将遭遇收敛性变慢或不收敛的问题。本项目首先利用一些稳定的有限元离散格式，例连续内点惩罚有限元方法(CIP)和间断Petrov-Galerkin 方法(DPG), 来构造有效的粗空间，对高波数Helmholtz方程和时谐Maxwell方程设计了非常有效的多重网格和区域分解格式，同时提出了一种Robin 型区域分解法可高效并行求解Helmholtz方程和时谐Maxwell方程，并从理论上严格证明了算法的收敛速度与波数无关。大量的数值实验验证了我们算法的有效性。在项目的资助下，发表了高水平论文11篇。