A-posteriori-steered $p$-robust multigrid with optimal step-sizes and adaptive number of smoothing steps - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles SIAM Journal on Scientific Computing Year : 2021

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

Abstract

We develop a multigrid solver steered by an a posteriori estimator of the algebraic error. We adopt the context of a second-order elliptic diffusion problem discretized by conforming finite elements of arbitrary polynomial degree $p ≥ 1$. Our solver employs zero pre- and one post-smoothing by the overlapping Schwarz (block-Jacobi) method and features an optimal choice of the step-sizes in the smoothing correction on each level by line search. This leads to a simple Pythagorean formula of the algebraic error in the next step in terms of the current error and level-wise and patch-wise error reductions. We show the two following results and their equivalence: the solver contracts the algebraic error independently of the polynomial degree $p$; and the estimator represents a two-sided $p$-robust bound on the algebraic error. The $p$-robustness results are obtained by carefully applying the results of Schöberl et al. [IMA J. Numer. Anal., 28 (2008), pp. 1–24] for one mesh, combined with a multilevel stable decomposition for piecewise affine polynomials of Xu et al. [Multiscale, nonlinear and adaptive approximation, Springer, Berlin, 2009, pp. 599–659]. We consider quasi-uniform or graded bisection simplicial meshes and prove mild dependence on the number of mesh levels for minimal $H^1$-regularity and complete independence for $H^2$-regularity. We also present a simple and effective way for the solver to adaptively choose the number of post-smoothing steps necessary at each individual level, yielding a yet improved error reduction. Numerical tests confirm $p$-robustness and show the benefits of the adaptive number of smoothing steps.
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Dates and versions

hal-02494538 , version 1 (28-02-2020)
hal-02494538 , version 2 (01-07-2020)
hal-02494538 , version 3 (14-10-2020)
hal-02494538 , version 4 (28-11-2020)
hal-02494538 , version 5 (22-11-2021)

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Cite

Ani Miraçi, Jan Papež, Martin Vohralík. A-posteriori-steered $p$-robust multigrid with optimal step-sizes and adaptive number of smoothing steps. SIAM Journal on Scientific Computing, 2021, Special Section: 2020 Copper Mountain Conference, 43 (5), pp.S117-S145. ⟨10.1137/20M1349503⟩. ⟨hal-02494538v5⟩
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