中文摘要：

基于有限元的渐近展开式，导出了新的外推公式，它们更精确地逼近密网上的有限元解（而不是微分方程的解）．提出了新的外推瀑布型多网格法（EXCMG），采用新外推公式及其二次插值提供密网上的好初值．数值实验表明，新方法有很高的精度和效率．最后在PC机上求解了大规模二维椭圆问题．

英文摘要：

Based on an asymptotic expansion of finite element solution, some new extrapolation formulas are derived, which are the exacter approximations to the finite element solution on refined grid （rather than the solution of partial differential equation）. A new extrapolation cascadic multigrid method（EXCMG） is proposed based on the better initial values on refined grid provided by the new extrapolation formulas and their quadratic interpolation. The numerical experiments show that the new algorithm is of the high accuracy and efficiency. Finally a large scale elliptic problem of second order is solved in PC.