A stopping criterion for the conjugate gradient algorithm in a finite element method framework
A stopping criterion for the conjugate gradient algorithm in a finite element method framework
复制标题
有限元方法框架中共轭梯度算法的停止准则
作者：
M. Arioli
摘要共轭梯度法一直被成功地用于求解通过有限元近似得到的对称和正定椭圆偏微分方程系统。最近的研究成果 [13, 19, 20, 22] 使得共轭梯度迭代过程中误差的能量规范得到了近似，考虑到这些研究成果，我们对 [3] 中引入的停止准则进行了调整。此外，我们还证明，使用高效的预处理器并不需要改变停止准则所使用的能量规范。最后，我们介绍了几个数值测试的结果，通过实验验证了我们的停止准则的有效性。
SummaryThe Conjugate Gradient method has always been successfully used in solving the symmetric and positive definite systems obtained by the finite element approximation of self-adjoint elliptic partial differential equations. Taking into account recent results [13, 19, 20, 22] which make it possible to approximate the energy norm of the error during the conjugate gradient iterative process, we adapt the stopping criterion introduced in [3]. Moreover, we show that the use of efficient preconditioners does not require to change the energy norm used by the stopping criterion. Finally, we present the results of several numerical tests that experimentally validate the effectiveness of our stopping criterion.