论文标题
相对于整体计算成本的自适应有限元方法的速率最佳性
Rate optimality of adaptive finite element methods with respect to the overall computational costs
论文作者
论文摘要
我们考虑用于二阶椭圆PDE的自适应有限元方法，其中未准确求解出现的离散系统。对于收缩迭代求解器，我们制定了一种自适应算法，该算法可以监视和引导自适应网格再填充以及出现的离散系统的不精确解决方案。我们证明，提出的策略会导致最佳代数速率线性收敛。但是，与先前的工作不同，我们将关注整体计算成本的收敛率。从明确的角度来看，提出的自适应策略可以保证准最佳计算时间。特别是，我们的分析涵盖了线性问题，其中线性系统通过最佳预处理的CG方法解决，以及具有强烈单调非线性的非线性问题，这些问题通过所谓的Zarantonello迭代线性化。
We consider adaptive finite element methods for second-order elliptic PDEs, where the arising discrete systems are not solved exactly. For contractive iterative solvers, we formulate an adaptive algorithm which monitors and steers the adaptive mesh-refinement as well as the inexact solution of the arising discrete systems. We prove that the proposed strategy leads to linear convergence with optimal algebraic rates. Unlike prior works, however, we focus on convergence rates with respect to the overall computational costs. In explicit terms, the proposed adaptive strategy thus guarantees quasi-optimal computational time. In particular, our analysis covers linear problems, where the linear systems are solved by an optimally preconditioned CG method as well as nonlinear problems with strongly monotone nonlinearity which are linearized by the so-called Zarantonello iteration.
