In this work, we study a local adaptive smoothing algorithm for a-posteriori-steered p -robust multigrid methods. The solver tackles a linear system which is generated by the discretization of a second-order elliptic diffusion problem using conforming finite elements of polynomial order p ≥ 1 {p\geq 1} . After one V-cycle (“full-smoothing” substep) of the solver of [A. Miraçi, J. Papež, and M. Vohralík, A-posteriori-steered p -robust multigrid with optimal step-sizes and adaptive number of smoothing steps, SIAM J. Sci. Comput. 2021, 10.1137/20M1349503], we dispose of a reliable, efficient, and localized estimation of the algebraic error. We use this existing result to develop our new adaptive algorithm: thanks to the information of the estimator and based on a bulk-chasing criterion, cf. [W. Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 1996, 3, 1106–1124], we mark patches of elements with increased estimated error on all levels. Then, we proceed by a modified and cheaper V-cycle (“adaptive-smoothing” substep), which only applies smoothing in the marked regions. The proposed adaptive multigrid solver picks autonomously and adaptively the optimal step-size per level as in our previous work but also the type of smoothing per level (weighted restricted additive or additive Schwarz) and concentrates smoothing to marked regions with high error. We prove that, under a numerical condition that we verify in the algorithm, each substep (full and adaptive) contracts the error p -robustly, which is confirmed by numerical experiments. Moreover, the proposed algorithm behaves numerically robustly with respect to the number of levels as well as to the diffusion coefficient jump for a uniformly-refined hierarchy of meshes.

中文翻译：

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基于A-Posteriori-steered p-Robust多重网格求解器的Dörfler标记的压缩局部自适应平滑

在这项工作中，我们研究了后向控制p鲁棒多重网格方法的局部自适应平滑算法。求解器使用多项式阶数p≥1 {p \ geq 1}的有限元来求解由二阶椭圆扩散问题离散化而生成的线性系统。在[A]的求解器的一个V循环（“全平滑”子步骤）之后。Miraçi，J。Papež和M.Vohralík，具有最佳步长和自适应步数的A后验转向p鲁棒多重网格，SIAM J. Sci。计算 2021，10.1137 / 20M1349503]，我们处理了可靠，高效且局部的代数误差估计。我们使用现有结果开发新的自适应算法：感谢估算器的信息，并基于批量购买标准cf。[W. 德福勒 泊松方程的收敛自适应算法，SIAM J. Numer。肛门 33 1996，3，1106–1124]，我们用所有级别的估计误差增加标记元素的补丁。然后，我们进行修改并更便宜的V循环（“自适应平滑”子步骤），该步骤仅在标记区域应用平滑处理。所提出的自适应多网格求解器可以像我们之前的工作那样自动地自适应选择每个级别的最佳步长，还可以选择每个级别的平滑类型（加权受限加性或加性Schwarz），并将平滑集中到具有高误差的标记区域。我们证明，在我们在算法中验证的数值条件下，每个子步骤（完全步骤和自适应步骤）均会稳健地收缩误差p，这已通过数值实验得到了证实。而且，