# A-posteriori-steered and adaptive $p$-robust multigrid solvers

Abstract : In this thesis, we consider systems of linear algebraic equations arising from discretizations of second-order elliptic partial differential equations using finite elements of arbitrary polynomial degree. In Chapter 1, we propose an a posteriori estimator for the algebraic error whose construction is inherently interconnected with the design of a multigrid solver with zero pre- and only one post-smoothing step by overlapping Schwarz (block-Jacobi) methods. The main contribution of this approach consists in the two following results and their equivalence: the solver contracts the algebraic error independently of the polynomial degree ($p$-robustness); the estimator represents a two-sided $p$-robust bound on the algebraic error. The proofs of these results hold in one, two, and three space dimensions, under the minimal $H^1$-regularity of the weak solution, for quasi-uniform meshes as well as for possibly highly graded ones, and are independent of the basis of the chosen finite element space. We introduce here an optimal step-size (by line search) in the error correction stage of the multigrid. In Chapter 2, we introduce level-wise optimal step-sizes, thus maximizing the decrease of the algebraic error on each level. Under the $H^2$-regularity assumption, we prove here that the $p$-robust contraction/efficiency also hold independently of the number of mesh levels. Apart from improving the performance of the solver, the use of the level-wise step-sizes also leads to an explicit Pythagorean formula of the decrease of the algebraic error from one iteration to the next. The formula then serves as foundation of a simple and effective adaptive strategy which allows the solver to choose the necessary number of post-smoothing steps on each level. In Chapter 3, we introduce an adaptive local smoothing strategy thanks to our efficient a posteriori estimator, which has the important property of being localized level-wise and patch-wise. Thus, the estimator can detect and mark which patches of elements among all mesh levels contribute more than a user prescribed percentage to the global algebraic error (via a bulk-chasing criterion). Each iteration of the adaptive solver is here composed of two sub-steps: after a first non-adaptive V-cycle, a second adaptive and inexpensive V-cycle employs local smoothing only in the marked patches. We prove that each of these sub-steps contracts the algebraic error $p$-robustly. Finally, in Chapter 4, we provide an extension of the above results to the case of the mixed finite element method discretization in two space dimensions. A variety of numerical tests is presented to confirm the theoretical findings of this thesis, as well as to show the benefits of our $p$-robust and/or adaptive solver approaches.
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Submitted on : Thursday, February 25, 2021 - 9:47:08 PM
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Ani Miraçi. A-posteriori-steered and adaptive $p$-robust multigrid solvers. Numerical Analysis [math.NA]. INRIA Paris; Sorbonne Universite, 2020. English. ⟨tel-03152913⟩

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