P意义下多重网格解法的收敛性

P有限元法是网格自适应过程中常被采用的一种增加逼近精度的办法.能在单元形状保持不变的情况下、通过提高插值多项式阶数而提高逼近精度.本文用泛函极小化序列的办法证明了P有限元方程多重网格解法的收敛性.收敛性定理证明,只要磨光过程中的松弛迭代是收敛的,P意义下的多重网格过程PMG(k, m, n, r)对于任何正整数m, n, r和k都是收敛的.文中还给出了求解Poisson方程及弹性力学方程的计算实例.算例表明,与P有限元法相结合的多重网格过程是一种有效的求解方法.

Convergence for P-Version of Multigrid Methods

The P-version finite element methods are usually applied for the adaptive mesh refinements. The improvement of the accuracy is obtained by increasing the order of the polynomial interpolation when the shape of the element does not change. In this paper the convergence for the P-version of the multigrid methods is proved using the minimal sequence of the functional. The conclusion of the convergence theorem is that for any positive integers m, n, r and k the procedure PMG(k, m, n, r) is convergent, provided the iteration scheme of the smoothing procedrue is convergent. Numerical examples for solving the poisson equation and the elastic mechanics problems are given. Numerical examples show that the multigrid procedure, with the P-version finite element is an effective algorithm.