由非残差估计器驱动的二阶线性椭圆型偏微分方程自适应有限元方法的最优复杂度，第一部分：对称偏微分方程
Optimal complexity of adaptive FEM for second-order linear elliptic PDEs driven by non-residual estimators, Part I: Symmetric PDEs
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中文总结 AI 辅助
研究对称二阶线性椭圆型偏微分方程的自适应有限元方法，在特定假设下证明算法能无条件全R线性收敛，对足够小参数保证最优复杂度，重点分析如基于平均的非残差估计器。
中文摘要 AI 辅助
我们考虑对称二阶线性椭圆型偏微分方程的自适应有限元方法，其中自适应算法控制局部网格细化以及迭代代数求解器。在对基础后验误差估计器和求解器的抽象假设下，我们证明通常的自适应算法导致无条件的全R线性收敛，与用户选择的自适应参数无关。对于足够小的参数，这保证了最优复杂度，即适当拟误差的衰减率相对于以通常非线性逼近类衡量的总体计算成本（进而时间）是最优的。与文献中的现有结果不同，主要关注对基于平均的估计器等非残差估计器的分析理解，如Zienkiewicz和Zhu提出的估计器或基于平衡通量的估计器。
英文摘要
We consider adaptive finite element methods for symmetric second-order linear elliptic PDEs, where the adaptive algorithm steers the local mesh refinement as well as an iterative algebraic solver. Under abstract assumptions on the underlying a-posteriori error estimator and the solver, we prove that the usual adaptive algorithm leads to unconditional full R-linear convergence, independently of the user-chosen adaptivity parameters. For sufficiently small parameters, this guarantees optimal complexity in the sense that the decay rate of an appropriate quasi-error is optimal with respect to the overall computation cost (and hence time) measured in terms of the usual nonlinear approximation classes. Unlike available results in the literature, the main focus is on the analytical understanding of non-residual estimators like averaging-based estimators as proposed by Zienkiewicz and Zhu or estimators based on equilibrated fluxes.
