高波数波动问题的快速算法研究

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 develop the stabilized discrete approaches for the time-harmonic Maxwell equation and the corresponding robust multilevel solvers in different frequency regimes.

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。

快速求解高波数Helmholtz方程和时谐Maxwell方程是工程计算领域非常重要且困难的问题，二者的快速算法研究在实际工程计算中有着非常广泛的运用。本项目完成的首要研究内容是设计一类稳定的并能减小污染误差的杂交间断Galerkin(HDG)算法来求解高波数Helmholtz方程；其次，对于高波数情形的Helmholtz方程，由于相应离散代数系统强不定，因此本项目基于稳定化的粗空间校正问题和粗空间中的稳定磨光算法，设计了具有鲁棒性的多水平快速求解器；另外，我们也进一步研究了基于一阶最小二乘格式的离散算法求解Helmholtz方程，并得到基于该算法连续解的稳定性估计和误差分析，当kh/p充分小且多项式次数p>O(log k)时算法可消除污染误差达到拟最优收敛。在高波数时谐Maxwell方程的研究中，我们研究了一类通过引入Lagrange乘子并基于混合型curl-curl系统的HDG算法，证明了该算法的无条件稳定性和相应的误差与波数、网格尺度、多项式次数的关系。