【作者】 陈鹏；

【导师】 舒适；

【作者基本信息】 湘潭大学 ， 计算数学， 2010， 硕士

【摘要】 针对两类晶格材料模型(参考节点分别为q=3和q=4)，我们设计了两种预处理方法．一种是以块对角逆为预条件子的共轭梯度法(BPCG)，另一种是以块下三角逆为预条件子的PGMRES方法．数值试验结果表明，BPCG法对q=3的情况有很好的求解效率和鲁棒性，但对q=4的情况，特别是当a很小时，其求解效率将变得很差．当a很小时，与BPCG法及以块对角逆为预条件子的PGMRES方法相比，以块下三角逆为预条件子的PGMRES方法对求解参考节点q=4的蜂窝状晶格离散模型在计算CPU和算法稳定性等方面均全面占优．在这两种预处理方法中，我们均利用了基于标量椭圆问题的GAMG法的PCG法求各个子块矩阵的逆，以提高内迭代运算效率；同时，通过对内迭代中预处理前、后备子块矩阵的条件数进行分析，表明利用GAMG法求解各个子方程组时，具有很好的求解效率和鲁棒性．另外，对q=3的情形，我们利用Taylor展开，建立了三种参考节点处的近似连续(PDEs)方程，通过对内迭代子系统对角子块矩阵的H10(Ω)椭圆性分析，为内迭代方法的合理性提供了部分理论支撑．更多还原

【Abstract】 In this paper, we propose two kinds of preconditioning methods for the discretemodels on lattice block materials with q = 3 and q = 4 reference nodes,respectively. One is the block preconditioned conjugate gradiet（BPCG） algorithmwith a block diagonal inverse as a preconditioner. The numerical resultshave been shown that the BPCG method is of good efficiency and robustness applyingto the solution of the lattice block material with q = 3, but for honeycomblattices （q = 4）, especially whenαis very small, the efficiency of this methodbecomes poor. To deal with such situations, we propose another preconditioningGMRES method （PGMRES） based on the block lower triangular inverse as a preconditioner.Whenαis very small, compared with the BPCG and the PGMRESwith a block diagonal preconditioner, this PGMRES method for the honeycomblattice（q = 4） discrete model is best both in the calculation of the stability andCPU times. In both preconditioning methods,in order to improve the efficiencyof the inner iterations, we have made use of PCG method with GAMG preconditionerbased on scalar elliptic problems to finding the inverse of each sub-blockmatrix. At the same time, internal iteration counts and the condition numbersof each sub-bloch matrix before and after preconditioning using GAMG methodare presented and it is shown that the PCG method with GAMG preconditionerhas better efficiency and robustness. In addition, for the lattice block materialwith q = 3, we use the Taylor’s expansion to establish the limiting continuous（PDEs） equations of three typical reference nodes. Theoretical support for thereasonableness of the inner iterations is partly presented by making the analysisof H₁⁰（Ω） elliptical condition for the sub-systems in the inner iterations.更多还原