2020
DOI: 10.1016/j.cma.2020.113243
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Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach. Recovering mass balance in any situation

Abstract: We present novel H(div) and H 1 liftings of given piecewise polynomials over a hierarchy of simplicial meshes, based on a global solve on the coarsest mesh and on local solves on patches of mesh elements around vertices on subsequent mesh levels. This in particular allows to lift a given algebraic residual. In connection with approaches lifting the total residual, we show how to obtain guaranteed, fully computable, and constant-free upper and lower a posteriori bounds on the algebraic, total, and discretizatio… Show more

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Cited by 10 publications
(33 citation statements)
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“…Recent developments dedicated a great deal of effort to account for inexactness of the algebraic approximations and introduce stopping criteria based on the interplay between discretization and algebraic computation in adaptive FEM. Among others, we mention the seminal contributions [10,26,4,5,34,30,33,32,21,31,29,20].…”
Section: Prolongate Smooth Solvementioning
confidence: 99%
“…Recent developments dedicated a great deal of effort to account for inexactness of the algebraic approximations and introduce stopping criteria based on the interplay between discretization and algebraic computation in adaptive FEM. Among others, we mention the seminal contributions [10,26,4,5,34,30,33,32,21,31,29,20].…”
Section: Prolongate Smooth Solvementioning
confidence: 99%
“…Another interesting approach consists in applying an adaptive construction of preconditioners, see, e.g., the recent work of Anciaux-Sedrakian et al [1], where the adaptivity relies on a posteriori error estimates of the algebraic error, cf. Papež et al [25,26], combined with a bulk-chasing criterion in the spirit of Dörfler [11]. To the best of the authors' knowledge, this is the first time a bulk-chasing criterion is used in an algebraic solver adaptivity (and not mesh refinement) setting.…”
Section: Introductionmentioning
confidence: 99%
“…However, these results still come with some limitations; namely additional iteration steps are required. This drawback was overcome by the estimates of [28] relying on a lifting of the algebraic residual that is carried out over a hierarchy of meshes with local, mutually independent discrete H(div, Ω) contributions. Here, a parallel with multigrid and domain decomposition techniques for H(div, Ω) relying on local problems, proposed in the work of Arnold et al [3,4], can be seen, where in [3] the authors construct a spectrally equivalent preconditioner for the operator I − grad div and in [4] multigrid solvers.…”
Section: Introductionmentioning
confidence: 99%
“…Here, a parallel with multigrid and domain decomposition techniques for H(div, Ω) relying on local problems, proposed in the work of Arnold et al [3,4], can be seen, where in [3] the authors construct a spectrally equivalent preconditioner for the operator I − grad div and in [4] multigrid solvers. From [28], guaranteed, fully computable, and efficient bounds on the algebraic, discretization, and total errors in conforming h and p finite element discretizations become available.…”
Section: Introductionmentioning
confidence: 99%