求解椭圆问题的一类外推三层网格法

摘要

The coarsest grid, coarse grid, finest grid and linear system of equations for two dimensional elliptic problem are given by apply linear Lagrange finite element, a better initial guess on coarse grid is given by using extrapolation formula (new extrapolation formula or classic extrapolation formula) on coarse grid and coarsest grid. Then a better initial value on finest grid is obtained by using cubic spline interpolation. A classic extrapolation three-level method and new extrapolation three-level method by combining with smooth operator are designed in this paper. Numerical experiment results show that the new methods are more efficient, comparing with usually cascadic multigrid method.%使用线性拉格朗日有限元离散一类二维椭圆问题,选择合适剖分尺度形成最粗网格、次粗网格和最细网格和对应的方程组.在最粗网格和次粗网格上使用外推法(新外推法或经典外推法)得到次粗网格上高精度近似解,然后使用三次样条插值为细网格提供初始值,结合磨光算子,构造了经典外推三层网格法和新外推三层网格法,并给出相应的数值实验.与通常的瀑布型多重网格法相比,数值实验表明了两种新算法计算精度更高,细层上迭代步数非常少,计算时间更短,具有较强的稳健性.