Current Problems in Numerical Analysis

A-posteriori-steered h- and p-robust multigrid solvers and adaptivity

Ani Miraçi

TU Wien

Friday, 9. December 2022 - 10:00 to 11:00

in IM, rear building, ground floor,

We study a symmetric second-order linear elliptic PDE discretized by piecewise polynomials of arbitrary degree p ≥ 1. To treat the arising linear system, we propose a geometric multigrid method with zero pre- and one post-smoothing step by an overlapping Schwarz (block-Jacobi) method [1]. Introducing optimal step sizes which minimize the algebraic error in the level-wise error correction step of multigrid, one obtains an explicit Pythagorean formula for the algebraic error. Importantly, this inherently induces a fully computable a posteriori estimator for the energy norm of the algebraic error. 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 carefully applying the results of [2] for one mesh, combined with a multilevel stable decomposition for piecewise affine polynomials of [3]. Two adaptive variants of this approach are also presented: a multigrid solver with an adaptive number of smoothing steps per level and a multigrid solver with adaptive local smoothing per patches. Moreover, recent developments in [4] allow to prove that a local variant of the solver is robust also with respect to the number of mesh levels which leads to an adaptive finite element method with optimal computational costs. Finally, we present a variety of numerical tests to confirm the theoretical results and to illustrate the advantages of our approach.

References:

[1] 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. 43, 5 (2021), S117–S145.

[2] J. Schöberl, J. M. Melenk, C. Pechstein, and S. Zaglmayr: Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements. IMA J. Numer. Anal., 28 (2008), pp. 1–24.

[3] L. Chen, R. H. Nochetto, and J. Xu: Optimal multilevel methods for graded bisection grids. Numer. Math., 120(1):1–34, 2012

[4] M. Innerberger, A. Miraçi, D. Praetorius, and J. Streitberger: Optimal computational costs of AFEM with optimal local hp-robust multigrid solver. Submitted.

References:

[1] 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. 43, 5 (2021), S117–S145.

[2] J. Schöberl, J. M. Melenk, C. Pechstein, and S. Zaglmayr: Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements. IMA J. Numer. Anal., 28 (2008), pp. 1–24.

[3] L. Chen, R. H. Nochetto, and J. Xu: Optimal multilevel methods for graded bisection grids. Numer. Math., 120(1):1–34, 2012

[4] M. Innerberger, A. Miraçi, D. Praetorius, and J. Streitberger: Optimal computational costs of AFEM with optimal local hp-robust multigrid solver. Submitted.