特殊鞍点问题的高效预处理技术及在电磁计算中的应用研究

项目名称： 特殊鞍点问题的高效预处理技术及在电磁计算中的应用研究

项目编号： No.11501525

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 张理涛

作者单位： 郑州航空工业管理学院

项目金额： 18万元

中文摘要： 工程和科学计算中的很多重要领域如计算电磁学、计算流体力学、偏微分方程离散、最优控制等,都涉及到大型稀疏鞍点问题的求解.由于鞍点问题的系数矩阵具有强不定性,通常需要对鞍点问题进行预处理,将其变为具有优良性质的等价线性方程组进行求解.本研究课题拟对特殊鞍点问题的高效预处理技术进行深入研究.一类是离散化混合型时谐Maxwell方程离散产生的鞍点问题;另一类是具有奇异(1,1)块的非对称鞍点问题.根据两类鞍点问题的结构特征和特殊性质,拟设计免增广和免Schur余的结构化预处理子,并对设计的预处理矩阵特征值分布、相应的特征向量、最小多项式和最优参数的选取进行理论分析,最后结合Krylov子空间方法,应用电磁计算中Oseen方程、Maxwell方程、Helmholtz方程和Stokes方程做数值试验,进一步验证和比较其性能.本研究课题拟设计的预处理子将对实际问题的求解提供重要的理论依据和应用价值.

中文关键词： 鞍点问题；Krylov子空间法；预处理技术；特征值分布；电磁计算

英文摘要： Many important areas of engineering and scientific computing such as computational fluid dynamics,computational electromagnetics, mixed finite element discretization of partial differential equations, optimal control, economics and so on come down to solving large sparse saddle point problems. Due to the strong uncertainty of the coefficient matrix of saddle point problems, it usually requires preconditioner into the equivalent linear equations with excellent nature. This research project is intended for in-depth study of the efficient preconditioning technology for special saddle point problems. One is the discreted mixed time-harmonic Maxwell equations which generate saddle point problems and the other is the nonsymmetric saddle point problems with singular (1,1) blocks. According to the structured characteristics and special nature of these saddle point problems, We plan to design augmentation-free and schur complement-free structured preconditioners and theoretically analyze eigenvalue distribution of preconditioning matrices, corresponding eigenvectors, minimal polynomial, and selection of the optimal parameters. Finally, We intend to combine Krylov subspace methods and apply Maxwell's equations、Oseen equations、Helmholtz equations as well as Stokes equations to do numerical experiments for further validating and comparing the performance of structured preconditioners. Preconditioners and efficient algorithms the project designed have important theoretical significance and practical value.

英文关键词： Saddle problem;Krylov subspace method;Preconditioning technique;Eigenvalue distribution;Electromagnetic computing