组合杂交四边形元的多重网格预处理共轭梯度方法

The Multigrid Preconditioned Conjugate Gradient Method for Combined Hybrid Quadrilateral Element

组合杂交元方法是一种求解弹性力学问题的稳定化有限元方法.为了快速求解组合杂交元离散得到的大型、稀疏、对称正定系统,本文研究了多重网格预处理共轭梯度方法.首先,通过选用合适的网格转移算子和光滑策略,得到了有效的多重网格预处理器.其次,通过分析数值试验结果证明所得到的多重网格预处理共轭梯度方法是有效可行的,利用该预处理方法大大降低了系数矩阵的条件数,提高了计算效率.此外,对于一类高性能的组合杂交元,多重网格预处理共轭梯度方法在网格畸变时依然收敛.

Combined hybrid finite elements method,applied to linear elasticity problem,is a stabilized finite element method.It is a large sparse symmetric positive definite system arising from combined hybrid quadrilateral element discretization.In order to solve the system quickly,the multigrid preconditioned conjugate gradient method(MGCGM)is introduced in this paper.Firstly,by choosing the appropriate intergrid transfer operators and smoothing strategy,an effective multigrid preconditioner is obtained.Then,numerical results show that MGCGM proposed is efficient,that is the condition number of the stiffness matrix is greatly decreased by the multigrid preconditioner.Moreover,the method still has a good convergence for the combined hybrid element with high performance in the case of the mesh distortion.