【作者】 舒适；

【导师】 许进超；

【作者基本信息】 湘潭大学 ， 计算数学， 2004， 博士

【摘要】 多重网格法是求解偏微分方程大规模离散化方程的最为有效的方法，粗略地讲，它可分为几何多重网格法和代数多重网格(AMG)法。这里，我们将结合几何和代数两种途径来研究多重网格法，并称之为所谓的基于几何和分析的代数多重网格法，这是目前国际上代数多重网格法研究领域中新发展起来的方法。 本文分为两部分。第一部分，结合几何和代数多重网格法的特点，我们为两类典型的复杂有限元方程组，设计了具有很强的Robust性和高效性的代数多重网格法。第一类是R~d，d=2，3中的高次Lagrange有限元方程组，其系数矩阵的具有较强稠密性：另一类是所谓的Criss-Cross网格下的线性有限元方程组的约化线性子系统，其相应的几何粗空间一般不具有嵌套性。通过对问题和有限元空间作深入、细致的分析，发现了许多重要的代数特征，形成了一些新的关于求解复杂有限元方程组的网格粗化技术和提升算子构造的代数方法，从本质上克服了粗网格层自由度难以控制等通常代数多重网格法的缺陷，该类AMG法还具有预处理(Setup)时间少、Robust性好和运算效率高等特性。进一步，通过引入新的证明方法，即利用所谓的Xu-Zikatanov恒等式等，我们从理论上严格证明了新算法的最优收敛(下降)率，数值试验验证了理论的正确性。另外，通过引入两套代数矩阵，对高次Lagrange有限元方程组，我们设计并分析了相应的基于AMG法的预条件共轭梯度法。这些主要的算法设计思想和理论分析方法，具有相当的普适性。 第二部分，我们针对两种应用问题，讨论和分析相应的代数多重网格法。第一种是晶格材料的离散模型。我们首先设计了一种基于AMG法的块预条件共轭梯度法，并就方形晶格模型，利用其近似连续模型，从理论上严格证明了其关于参数α是一致收敛性。接着又构造了对更广泛的晶格模型具有高效性和Robust性的AMG法和相应的APCG法，数值试验表明我们的算法对许多晶格模型，关于其规模和重要参数α是一致收敛的。第二种应用问题来源于辐射流体力学方程组，我们讨论其中的二维三温能量方程离散系统的代数多重网格法。我们针对二维三温能量方程的特殊性，建立了一种半粗化的代数多重网格法(SAMG)和以该SAMG为预条件子的Krylov子空间迭代法，并将其嵌入到能量方程与流体力学方程耦合后得到的应用程序中，通过与经典预条件GEMRES(m)和ORTHOMIN(m)迭代法作对比数值实验，表明我们的AMG方法具有高效性和很好的Robust性。更多还原

【Abstract】 Multigrid methods are by far the most efficient methods for solving large scale algebraic systems arising from discretizations of partial differential equations. Roughly speaking, these methods can be developed though two approaches: geometric approach and algebraic approach. The purpose of this work is to develop multigrid methods by combining both geometric and algebraic approaches, which may be known as algebraic multigrid(AMG) methods based on geometric and analytic information.This dissertation consists of two parts. In the first part, by combining advantages of both geometric and algebraic multigrid methods, some robust multigrid methods are constructed for two kinds of finite element equations, one is the high order Lagrangian finite element equation for which AMG is often not very efficient; another is a condensed finite element system on criss-cross grids where the corresponding coarse spaces can not be made nested easily. By an effective use of geometric information and analytic properties of underlying differential equations and finite element spaces, many algebraic features can be exploited for developing new coarsening techniques and new interpolation operators. The new method overcomes the difficulty for properly controlling the degrees of freedom of the coarse spaces in the usual AMG methods. Numerical results show that our algebraic multigrid algorithm is substantially better than many usual algebraic multigrid algorithms. Furthermore, by using a new theoretical approach, namely the Xu-Zikatanov identity, a rigorous convergence analysis of our algebraic multigrid method is given. In addition, based on an algebraic multigrid method of linear finite elements, a robust preconditioned conjugate gradient method is presented and analyzed for the discrete systems of high order Lagrangian finite elements. These new ideas of hybrid multigrid and theoretical analysis provided in this work can be extended to more general cases.In the second part, algebraic multigrid methods are applied to solve two kinds of discrete systems arising from practical applications. First a block preconditioned conjugate gradient method (BPCG) and a class of algebraic multigrid methods are developed and studied for some discrete mathematical models for lattice block materials. Numerical experiments show that the new AMG methods converge uniformly with respect to the size of problem and also to some crucial parameters. Such a uniform convergence of the BPCG algorithm is further theoretically justified properly by更多还原