Adaptive inexact iterative algorithms based on polynomial-degree-robust a posteriori estimates for the Stokes problem

  • Martin Čermák
  • Frédéric Hecht
  • Zuqi Tang
  • Martin Vohralík
Article

Abstract

In this paper, we develop adaptive inexact versions of iterative algorithms applied to finite element discretizations of the linear Stokes problem. We base our developments on an equilibrated stress a posteriori error estimate distinguishing the different error components, namely the discretization error component, the (inner) algebraic solver error component, and possibly the outer algebraic solver error component for algorithms of the Uzawa type. We prove that our estimate gives a guaranteed upper bound on the total error, as well as a polynomial-degree-robust local efficiency, and this on each step of the employed iterative algorithm. Our adaptive algorithms stop the iterations when the corresponding error components do not have a significant influence on the total error. The developed framework covers all standard conforming and conforming stabilized finite element methods on simplicial and rectangular parallelepipeds meshes in two or three space dimensions and an arbitrary algebraic solver. Implementation into the FreeFem++ programming language is invoked and numerical examples showcase the performance of our a posteriori estimates and of the proposed adaptive strategies. As example, we choose here the unpreconditioned and preconditioned Uzawa algorithm and the preconditioned minimum residual algorithm, in combination with the Taylor–Hood discretization.

Keywords

Stokes problem Conforming finite element method Adaptive inexact iterative algorithm Outer-inner iteration Uzawa method MinRes A posteriori error estimate Guaranteed bound Efficiency Polynomial-degree-robustness Interplay between error components Adaptive stopping criterion 

Mathematics Subject Classification

65N15 65N22 65N30 65F10 76M10 

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Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  • Martin Čermák
    • 1
  • Frédéric Hecht
    • 2
    • 3
    • 4
  • Zuqi Tang
    • 5
  • Martin Vohralík
    • 4
    • 6
  1. 1.IT4Innovations National Supercomputing Center (IT4I)VSB - Technical University of OstravaPoruba, OstravaCzech Republic
  2. 2.Laboratoire Jacques-Louis Lions, UMR 7598UPMC Univ Paris 06ParisFrance
  3. 3.Laboratoire Jacques-Louis Lions, UMR 7598CNRSParisFrance
  4. 4.INRIA ParisParis 12France
  5. 5.Univ. Lille, Centrale Lille, Arts et Métiers Paris Tech, HEIEA 2697 - L2EP - Laboratoire d’Electrotechnique et d’Electronique de PuissanceLilleFrance
  6. 6.Université Paris-EstCERMICS (ENPC)Marne-la-Vallée 2France

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