几类面向PDEs问题特征的高效并行多层迭代法研究

多层迭代法是国际上流行的求解偏微分方程大规模离散系统的快速算法和计算数学的研究热点。代数多重网格（AMG）法是多层迭代法的典型代表和计算软件平台的首选多重网格解法器。基于约束能量极小粗空间的区域分解法（CEM-DDM）是近年发展起来的另一类重要多层迭代法。基于问题（或应用）特征的多层迭代法是提高算法效率的自然选择。本项目针对三维对流扩散等模型方程的非对称离散系统，重点研究非对称AMG法；同时针对具有典型应用特征的三维非结构网格下辐射扩散问题和三维高波数问题，研究高效自适应AMG法和CEM-DDM及其并行解法器，其中涉及许多关键难点（特别是“强多尺度”、“非对称”、“耦合的多样化”和“高度病态”等特征带来的影响），需要发展一些新方法、新技术和新途径。本课题既是多层迭代法的基础创新算法和理论研究，更是提升实际应用问题计算效率的迫切需要，所获成果还可为具有这些典型特征的其它应用问题提供借鉴。

Multilevel iterative methods are internationally popular fast algorithms for solving large-scale discretized systems of partial differential equations and are also hot research topics in computational mathematics. Algebraic multigrid method (AMG) is a typical representative and the preferred multigrid solver for computing software platforms. Domain decomposition method based on constrained energy minimizing coarse spaces (CEM-DDM) is another important multilevel iterative method developed in recent years. The multilevel iterative method based on problem or application features is a natural choice to improve the efficiency of the algorithm. This project aims at the nonsymmetric discretized systems of three-dimensional convection-diffusion model equation, with the emphasis on nonsymmetric AMG methods. Meanwhile, efficient adaptive AMG methods and CEM-DDM as well as their parallel solvers are constructed for radiation diffusion equations on three-dimensional unstructured grids and three-dimensional equations with high wave number, where involve numerous difficulties, such as the impacts of the spatial and temporal multiscale nature, nonsymmetry, diversification of couplings and highly ill-conditioned systems. Some novel ways and new techniques are needed to dispose of those difficulties. This project is a fundamental issue to innovative algorithms and theoretical researches of multilevel iterative methods, and moreover, is an urgent need to improve the computational efficiency of practical application problems. The achievements of the project will provide an important reference to other application problem with these typical features.