高波数波动问题的快速算法研究

项目名称： 高波数波动问题的快速算法研究

项目编号： No.11201394

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 陈黄鑫

作者单位： 厦门大学

项目金额： 22万元

中文摘要： Helmholtz方程和时谐Maxwell方程是波动问题中两个重要而基本的方程，在实际物理问题和工程计算中有着非常广泛的运用，二者的快速算法研究将会推动许多相关领域的发展。在高波数问题中，由于数值耗散的影响，采用传统方法求解会产生较大的污染误差；其次，采用通常的迭代算法求解对应的离散问题收敛速度很慢。本项目将对Helmholtz方程系统研究其稳定的离散方法以减弱污染误差的影响，着重考虑一类可杂交化的间断Galerkin方法；深入研究求解高波数Helmholtz方程高效稳定的多水平方法，结合稳定的离散方法设计有效的粗空间校正问题和粗细网格上稳定的磨光算法，分析预处理系统的谱分布；在基于前述Helmholtz方程研究的基础上，结合棱有限元进一步探讨时谐Maxwell方程稳定的离散方法，研究不同波数情形对应的多水平求解器。

中文关键词： Helmholtz方程；高波数；间断Galerkin方法；多水平方法；时谐Maxwell方程

英文摘要： Helmholtz equation and time-harmonic Maxwell equation, which play key roles in physics and engineering, are two important and fundamental equations in the wave problems. The development of fast methods for the two equations is very critical in many practical applications. For the problems with high wave numbers, due to the numerical dispersion the phase errors tend to accumulate and induce the so-called pollution error, which is especially inherent in the standard numerical methods. Moreover, when the standard iterative methods are applied to solve the discrete problem, the convergence rates are usually quite slow. In this research project we will study the stabilized discrete approaches for the Helmholtz equation in order to reduce the pollution error. Especially, we will focus on the hybridizable discontinuous Galerkin method. We also aim to develop the robust multilevel methods for the Helmholtz equation with high wave numbers. On one hand we will apply the stabilized discrete approaches to design efficient correction problems on the coarse grids and robust smoothers on the fine and coarse grids. On the other hand, the spectral distribution of the preconditioning system also needs to be further considered. Based on the works for the Helmholtz equation, we will combine the edge finite element approximation to

英文关键词： Helmholtz equation；High wave numbers；Discontinuous Galerkin method；Multilevel methods；Time-Harmonic Maxwell equation