基于HSS 型光滑子的多重网格方法

It is well known that multigrid method is optimal for solving the discrete equations which arise in numerical approximation of elliptic boundary value problems. This program is devoted to study the multigrid method with HSS-type smoothers. HSS iteration method and the modified HSS iteration method converge unconditionally to the unique solution of the non-Hermitian positive definite system of linear equations. Moreover, these methods have the same convergence rate as the conjugate gradient method. Especially for solving the complex symmetric systems of linear equations, each iteration of the modified HSS iteration method only requires to solve two linear systems with real symmetric positive definite coefficient matrices. In this program, based on the new perturbation argument technique, we will study the convergence of the multigrid methods with HSS-type smoothers for solving complex coefficient elliptic problems ( the dominant coefficient is complex valued non-vanishing function)，and give some numerical examples to show the efficiency of these methods. Because of the properties of the HSS-type methods and the optimality of the multigrid method, the research of multigrid methods with HSS-type smoothers has much significance both theoretically and practicaly.

对于求解由有限元方法离散椭圆边值问题得到的线性方程组，多重网格方法是一种最优的数值方法。本项目将研究 HSS 型迭代法做光滑子的多重网格方法。作为一类新近提出的数值方法, HSS 迭代法及修改的 HSS 迭代法对于求解 non-Hermitian 正定线性方程组是无条件收敛的，且其收敛速度与共轭梯度法相当。特别对于求解复对称正定线性方程组，修改的 HSS 迭代法在迭代过程中只需要求解相应的实对称正定线性方程组。因此，HSS 型迭代法在理论和数值上都具有很好的结果。本项目将以复系数二阶椭圆方程（高阶项的系数为复值函数）为模型，研究 HSS 型迭代法做光滑子的多重网格方法，基于新提出的扰动论证技巧，给出该类方法的收敛性分析，并通过数值实验验证其有效性。由于 HSS 型迭代法的上述特点和多重网格方法的最优性， 所以研究基于 HSS 型光滑子的多重网格方法具有重要的理论意义和应用价值。

本项目主要研究基于 HSS 型光滑子的多重网格算法求解复系数椭圆方程（高阶项的系数为复值函数）。在理论上，我们将求解实系数椭圆方程的多重网格算法理论推广应用到了求解复系数椭圆方程上，给出了基于 PMHSS 光滑子的多重网格算法的收敛性分析，证明了该算法的最优收敛性，即其收敛率与网格步长无关。在数值上，我们分别给出了基于 HSS, MHSS 和 PMHSS 光滑子的多重网格算法求解复系数椭圆方程的数值结果。并与基于 Gauss-Seidel光滑子的多重网格算法做了比较，数值结果表明新提出算法的收敛性对方程的系数变化依赖性较小。因此本项目在理论和数值上都验证了所提出多重网格算法的最优性和有效性，也为进一步研究多重网格方法求解其它复系数方程提供了研究基础。