A posteriori-steered $p$-robust multigrid with optimal step-sizes and adaptive number of smoothing steps

Ani Miraçi (1, 2) , Jan Papež (1, 2) , Martin Vohralík (1, 2)

1 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

In this work, we develop a multigrid solver that is steered by a posteriori estimates of the algebraic error. We adopt the context of a second order elliptic diffusion problem discretized by the conforming finite element method of arbitrary polynomial degree p ≥ 1. Our solver in particular features an optimal (adaptive) choice of the step-size in the smoothing correction on each level. Developing our previous work [HAL Preprint 02070981, 2019], we show the two following results and their equivalence: 1) the solver contracts the algebraic error independently of the polynomial degree p; 2) the estimator represents a two-sided p-robust bound on the algebraic error. The p-robustness results are obtained by careful application of the work done in Schöberl et al. [IMA J. Numer. Anal., 28 (2008), pp. 1-24] for one given mesh, combined with a multilevel stable decomposition for piecewise affine polynomials on quasi-uniform/bisection grids given in Xu et al. [Springer, Berlin, 2009, pp. 599-659]. We consider either quasi-uniform or graded bisection meshes and show robustness with respect to the number of mesh levels J for H 2-regular solutions. Our solver relies on zero pre-and one post-smoothing by an overlapping Schwarz (block-Jacobi) method. We also present a simple and effective way for the solver to adaptively choose the number of post-smoothing steps, which yields yet improved error reduction. We present numerical tests confirming the p-robust behavior of the solver and illustrating the adaptive number of smoothing steps. Moreover, the tests indicate numerical robustness with respect to the number of levels J even in low regularity settings, as well as robustness with respect to the jumps in diffusion coefficient.

Mots clés: Finite element method; Multigrid method; Schwarz method; Block-Jacobi smoother; A posteriori estimate; Stable decomposition; p-robustness

Domaines: Analyse numérique [math.NA]

Ani Miraçi, Jan Papež, Martin Vohralík. A posteriori-steered $p$-robust multigrid with optimal step-sizes and adaptive number of smoothing steps. 2020. ⟨hal-02494538v1⟩