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# Guaranteed contraction of adaptive inexact $hp$-refinement strategies with realistic stopping criteria

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Abstract : The purpose of this contribution it to theoretically analyze the adaptive refinement strategies for conforming hp-finite element approximations of elliptic problems proposed for exact algebraic solvers in [Daniel, Ern, Smears, Vohralík, Comput. Math. Appl. 76 (2018), 967-983] and for inexact algebraic solvers in [Daniel, Ern, Vohralík, Comput. Methods Appl. Mech. Engrg. 359 (2020), 112607]. Both of these strategies are driven by guaranteed equilibrated flux energy error estimators. The employed hp-refinement criterion stems from solving two separate local residual problems posed only on the patches of elements around marked vertices, selected by a bulk-chasing criterion. In the above references, we have derived a fully computable guaranteed bound on the ratio of the error on two successive steps of the hp-adaptive loop. Here, our focus is to prove that this ratio is uniformly smaller than one, which implies a guaranteed contraction of the adaptive and adaptive inexact hp-refinement strategies. To be able to achieve this goal, we have to introduce some additional assumptions on the h and p-refinements, namely the so-called interior node property, an extension of the marked region, and a sufficient polynomial degree increase. In the inexact case, a sufficiently precise stopping criterion for the algebraic solver is requested, but this criterion remains fully computable and also realistic in the sense that in numerical experiments, it does not request the algebraic error to be excessively small in comparison with the total error.
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https://hal.inria.fr/hal-02486433
Contributor : Martin Vohralik <>
Submitted on : Thursday, February 20, 2020 - 11:53:30 PM
Last modification on : Monday, November 16, 2020 - 11:46:10 AM
Long-term archiving on: : Thursday, May 21, 2020 - 5:38:18 PM

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• HAL Id : hal-02486433, version 1

### Citation

Patrik Daniel, Martin Vohralík. Guaranteed contraction of adaptive inexact $hp$-refinement strategies with realistic stopping criteria. 2020. ⟨hal-02486433⟩

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