An algebraic theory for multigrid methods for variational problems
An algebraic theory for multigrid methods for variational problems
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DOI：
10.1137/0725008
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发表时间：
1988-02
影响因子：
2.9
通讯作者：
J. Mandel;S. McCormick;J. Ruge
中科院分区：
文献类型：
--
作者：
J. Mandel;S. McCormick;J. Ruge
发展了变分环境下对称正定问题多重网格方法的收敛理论。该理论是基于一个单一的代数近似假设，这是满足有限元离散的椭圆边值问题，虽然该理论可以适用于问题没有任何连续的背景，以及。与以前的结果相比，我们证明了快速收敛与任何正数的平滑步骤的V-和W-循环下的离散类似的$H ^2 $和$H^{1 + \alpha } $正则性假设，分别。我们分析了一个广泛的类平滑，包括任意对称和非对称的预处理迭代，任意顺序的高斯-赛德尔，最速下降，Chebyshev迭代和共轭梯度。我们的估计表现出通常的渐近行为，大量的平滑步骤。
A convergence theory is developed for multigrid methods for symmetric, positive definite problems in a variational setting. The theory is based on a single algebraic approximation assumption, which is satisfied for finite element discretizations of elliptic boundary value problems, although the theory can be applied to problems without any continuous background as well. In contrast to previous results, we prove fast convergence with any positive number of smoothing steps for V- and W-cycles under discrete analogues of the $H^2 $ and $H^{1 + \alpha } $ regularity assumptions, respectively. We analyze a wide class of smoothers, including arbitrary symmetric and nonsymmetric preconditioned iterations, arbitrarily ordered Gauss–Seidel, steepest descent, Chebyshev iteration and conjugate gradients. Our estimates exhibit the usual asymptotic behavior for a large number of smoothing steps.