中文摘要：

我们在场并且分析柔韧的 preconditioned 为椭圆形的问题的一个班的更高的顺序 Lagrangian 有限元素系统结合坡度方法。一个辅助线性元素僵硬矩阵被选择是为更高的顺序的 preconditioner 有限元素。然后一代数学多，线性有限元素的格子方法被申请解决 preconditioner。独立于网孔尺寸的最佳的条件数字被获得。数字实验证实算法的效率。

英文摘要：

We present and analyze a robust preconditioned conjugate gradient method for the higher order Lagrangian finite element systems of a class of elliptic problems. An auxiliary linear element stiffness matrix is chosen to be the preconditioner for higher order finite elements. Then an algebraic multigrid method of linear finite element is applied for solving the preconditioner. The optimal condition number which is independent of the mesh size is obtained. Numerical experiments confirm the efficiency of the algorithm.