对流扩散方程有限元逼近的预处理HSS方法
Preconditioned HSS Method for Finite Element Approximations of Convection-Diffusion Equations
AI总结：
针对对流扩散方程有限元离散的非对称线性系统，提出基于Hermitian/斜Hermitian分裂的两步预处理迭代法，通过Toeplitz理论证明均匀网格下预处理矩阵序列强聚簇于1，实现超线性预处理，并验证最优性。
摘要：
基于Hermitian/斜Hermitian分裂的两步预处理迭代法被应用于求解由对流扩散方程有限元逼近产生的非对称线性系统。理论谱分析聚焦于与均匀结构化网格上的有限元逼近相关的矩阵序列情形，借助Toeplitz理论导出的谱工具。在此设定下，若问题具有强制性，且扩散和对流系数足够正则，则所提出的预处理矩阵序列在单位元处呈现强聚簇，即获得超线性预处理序列。在相同假设下，证明了PHSS方法的最优性，数值实验证实了理论结果。还给出了非结构化网格上的测试，显示了类似的收敛行为。
英文摘要：
A two-step preconditioned iterative method based on the Hermitian/Skew-Hermitian splitting is applied to the solution of nonsymmetric linear systems arising from the Finite Element approximation of convection-diffusion equations. The theoretical spectral analysis focuses on the case of matrix sequences related to FE approximations on uniform structured meshes, by referring to spectral tools derived from Toeplitz theory. In such a setting, if the problem is coercive, and the diffusive and convective coefficients are regular enough, then the proposed preconditioned matrix sequence shows a strong clustering at unity, i.e., a superlinear preconditioning sequence is obtained. Under the same assumptions, the optimality of the PHSS method is proved and some numerical experiments confirm the theoretical results. Tests on unstructured meshes are also presented, showing the some convergence behavior.
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