Contractive Local Adaptive Smoothing Based on Dörfler’s Marking in A-Posteriori-Steered p-Robust Multigrid Solvers

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.

Numerical methods for black box software

Сформулированы требования к вычислительным алгоритмам для перспективного программного обеспечения, устроенного по принципу "черного ящика" и предназначенного для математического моделирования в механике сплошных сред. Выполнен анализ прикладных свойств классических многосеточных методов и универсальной многосеточной технологии в рамках проблемы "универсальность-эффективность-параллелизм". Показано, что близкая к оптимальной трудоемкость при минимуме проблемно-зависимых компонентов и высокая эффективность параллелизма достижимы при использовании универсальной многосеточной технологии на глобально структурированных сетках. Применение неструктурированных сеток потребует определения двух проблемно-зависимых компонентов (межсеточных операторов), которые значительно влияют на трудоемкость алгоритма. A number of requirements are formulated to the numerical algorithms for black box software intended for mathematical modeling in continuum mechanics. An analysis of applied properties of the classical multigrid methods and robust multigrid technique in the framework of "robustness-efficiency-parallelism" problem is performed. It is shown that a close-to-optimal complexity with the least number of problem-dependent components and high parallel efficiency can be achieved with the robust multigrid technique on globally structured grids. Application of unstructured grids requires the accurate definition of two problem-dependent components (intergrid operators) that strongly affect on the complexity of an algorithm.