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Residual-based a posteriori error estimation for hp-adaptive finite element methods for the Stokes equations

Arezou Ghesmati, Wolfgang Bangerth and Bruno Turcksin


We derive a residual-based a posteriori error estimator for the conforming hp-Adaptive Finite Element Method (hp-AFEM) for the steady state Stokes problem describing the slow motion of an incompressible fluid. This error estimator is obtained by extending the idea of a posteriori error estimation for the classical h-version of AFEM. We also establish the reliability and efficiency of the error estimator. The proofs are based on the well-known Clément-type interpolation operator introduced in [27] in the context of the hp-AFEM. Numerical experiments show the performance of an adaptive hp-FEM algorithm using the proposed a posteriori error estimator.

JEL Classification: 65N30; 65N15


This material is based upon work supported by the U.S. Department of Energy, Office of Science, under contract number DE-AC05-00OR22725. AG and WB’s work was supported by the National Science Foundation under award OCI−1148116 as part of the Software Infrastructure for Sustained Innovation (SI2) program. WB was also supported by the Computational Infrastructure in Geodynamics initiative (CIG), through the National Science Foundation under Awards No. EAR-0949446 and EAR−1550901, administered by The University of California, Davis.

This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (


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Received: 2018-04-20
Revised: 2018-10-14
Accepted: 2018-10-27
Published Online: 2018-11-02
Published in Print: 2019-12-18

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