基于矩形有限元离散泊松方程的二重网格法研究

Research on the Double Grid Method Based on the Rectangular Finite Element Discrete Poisson Equation

采用矩形有限元方法离散泊松方程,形成粗网格和细网格,对粗网格精确求解,然后采用三次样条插值为细层提供初始值,构造出一类求解二维泊松方程的瀑布型二重网格法,并结合数值实验讨论不同粗细网格情况下算法的计算效率。结果表明,只需要使用节点较少的粗网格插值算子、磨光算子就能有效求出节点较多细网格上的数值解,改进的算法耗时更短。

The rectangular finite element method is used to discretize the Poisson equation to form a coarse grid and a fine grid.The coarse grid is accurately solved,and then the cubic spline interpolation is used to provide the initial value for the fine layer,and a class of solving the two-dimensional Poisson equation is constructed.The waterfall-type double-grid method is combined with numerical experiments to discuss the computational efficiency of the algorithm under different grid thicknesses.The results show that only the coarse grid interpolation operator and the polishing operator with fewer nodes can effectively obtain the numerical solution on the fine grid with more nodes,and the time-consuming improvement of the algorithm is shorter.