求解二阶椭圆奇性问题的组合网格法

Composite grid method for solving singularity problems of second order elliptic equation

针对科学与工程计算中改进奇异问题的计算精度,提出了一种新的算法—组合网格法(CGM)。即全局粗网格与局部细网格之间交替迭代。所谓组合网格由整体粗网格和局部加密细网格组成。通过计算表明,组合网格方法和快速自适应组合网格方法(FAC)一样快速收敛。但本质不同的是并不要求粗细网格嵌套。而且还能做到总体与局部的算子不一样,与普通有限元方法比较,该方法形成的线性方程组的系数矩阵的阶数要小得多,避免了求解大型矩阵方程。但计算结果接近全局加密网格上的结果,更适用工程实际问题。

In order to solve singularity problems occurring in engineering and scientific computation and to increase the accuracy of the approximate solution of singularity problems, a new algorithm-composite grid method （CGM） is proposed in this paper. Specifically, we consider alternative iteration between global coarse grids and local fine grids. The composite grid consists of a global coarse grid and a fine local grid. The calculation results indicate that composite grid method has fast convergence, which is the same as the fast adaptive composite grid method （FAC）. However, the coarse grid and the fine grid do not require to be nonnested. In addition, the global operators and lo- cal operators can be different. The comparison of the method with ordinary finite element shows that with the same order accuracy, the CGM method requires less computer storage and avoids the solving of large matrix. This method provides a new way for engineering and scientific computation.