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Journal of Numerical Mathematics

Editor-in-Chief: Hoppe, Ronald H. W. / Kuznetsov, Yuri

Managing Editor: Olshanskii, Maxim

Editorial Board: Benzi, Michele / Brenner, Susanne C. / Carstensen, Carsten / Dryja, M. / Feistauer, Miloslav / Glowinski, R. / Lazarov, Raytcho / Nataf, Frédéric / Neittaanmaki, P. / Bonito, Andrea / Quarteroni, Alfio / Guzman, Johnny / Rannacher, Rolf / Repin, Sergey I. / Shi, Zhong-ci / Tyrtyshnikov, Eugene E. / Zou, Jun / Simoncini, Valeria / Reusken, Arnold


IMPACT FACTOR 2018: 3.107

CiteScore 2018: 2.43

SCImago Journal Rank (SJR) 2018: 1.252
Source Normalized Impact per Paper (SNIP) 2018: 1.618

Mathematical Citation Quotient (MCQ) 2018: 1.13

Online
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1569-3953
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Volume 22, Issue 3

Issues

A non-conforming composite quadrilateral finite element pair for feedback stabilization of the Stokes equations

P. Benner
  • Research Group Computational Methods in Systems and Control Theory (CSC), Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg, Sandtorstr. 1, 39106 Magdeburg, Germany/Research Group Mathematics in Industry and Technology (MiIT), Chemnitz University of Technology, Reichenhainer Str. 39/41, 09126 Chemnitz, Germany
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ J. Saak
  • Research Group Computational Methods in Systems and Control Theory (CSC), Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg, Sandtorstr. 1, 39106 Magdeburg, Germany/Research Group Mathematics in Industry and Technology (MiIT), Chemnitz University of Technology, Reichenhainer Str. 39/41, 09126 Chemnitz, Germany
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ F. Schieweck
  • Institute for Analysis und Numerics, Otto-von-Guericke-University of Magdeburg, Postfach 4120, 39016 Magdeburg, Germany
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ P. Skrzypacz
  • Research Group Computational Methods in Systems and Control Theory (CSC), Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg, Sandtorstr. 1, 39106 Magdeburg, Germany
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ H.K. Weichelt
  • Research Group Mathematics in Industry and Technology (MiIT), Chemnitz University of Technology, Reichenhainer Str. 39/41, 09126 Chemnitz, Germany
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2014-10-07 | DOI: https://doi.org/10.1515/jnma-2014-0009

Abstract

- In this contribution, we show a method for the boundary feedback stabilization of the Stokes problem around a stationary trajectory. We derive a formal low-rank algorithm for solving the stabilization problem in operator notation. The appearing operator equations are formulated in terms of stationary partial differential equations (PDEs) instead of using their finite dimensional representations in terms of matrices. A Galerkin method, satisfying the divergence constraint pointwise locally is especially appealing since it represents appropriately the action of the Helmholtz projection.

The main advantages of the composite technique are the efficient assembly of element matrices, the reduction of computational costs using static condensation, and the diagonal mass matrix. The non-conforming character of the composite element guarantees a better sparsity pattern, compared to conforming elements, due to the lower number of couplings between basis functions corresponding to neighboring cells. We also achieve the pointwise mass conservation on sub-triangles of each element.

Keywords: non-conforming finite elements; Stokes equations; feedback stabilization

About the article

Published Online: 2014-10-07

Published in Print: 2014-10-01


Citation Information: Journal of Numerical Mathematics, Volume 22, Issue 3, Pages 191–220, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI: https://doi.org/10.1515/jnma-2014-0009.

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