The sub-Cauchy–Stokes problem: Solvability issues and Lagrange multiplier methods with artificial boundary conditions

The sub-Cauchy–Stokes problem: Solvability issues and Lagrange multiplier methods with... In contrast to the conventional Cauchy–Stokes problems, in which the velocity and the stress force data are given on the accessible boundary, in the present paper, we reduce the accessible boundary data information and we consider a problem which deals only with shear stress data. We refer to this problem as a sub-Cauchy–Stokes problem. This problem is ill-posed because of severe instability and even uniqueness is unknown. We first address the uniqueness issues associated with this problem. Resorting to the domain decomposition techniques together with the duplication process of Vogelius (Kohn and Vogelius, 1985), we propose new Lagrange multiplier methods to solve the sub-Cauchy–Stokes problem. These methods consist in recasting the problem in terms of interfacial equations, by equalizing two solutions of the sub-Cauchy–Stokes problem using matching conditions defined on the inaccessible boundary. The matching is based on second order conditions and the types of the interfacial equations depend on the equations used to match the values of the unknowns on the inaccessible boundary. The interfacial problems are then solved by iterative procedures in which coefficients can be optimized to improve convergence rates. A complete analysis of the methods is presented, and intensive numerical results illustrate the effectiveness and the performance of the proposed approaches. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Computational and Applied Mathematics Elsevier

The sub-Cauchy–Stokes problem: Solvability issues and Lagrange multiplier methods with artificial boundary conditions

Loading next page...
 
/lp/elsevier/the-sub-cauchy-stokes-problem-solvability-issues-and-lagrange-CNLjbRj4i0
Publisher
Elsevier
Copyright
Copyright © 2018 Elsevier B.V.
ISSN
0377-0427
eISSN
1879-1778
D.O.I.
10.1016/j.cam.2018.01.034
Publisher site
See Article on Publisher Site

Abstract

In contrast to the conventional Cauchy–Stokes problems, in which the velocity and the stress force data are given on the accessible boundary, in the present paper, we reduce the accessible boundary data information and we consider a problem which deals only with shear stress data. We refer to this problem as a sub-Cauchy–Stokes problem. This problem is ill-posed because of severe instability and even uniqueness is unknown. We first address the uniqueness issues associated with this problem. Resorting to the domain decomposition techniques together with the duplication process of Vogelius (Kohn and Vogelius, 1985), we propose new Lagrange multiplier methods to solve the sub-Cauchy–Stokes problem. These methods consist in recasting the problem in terms of interfacial equations, by equalizing two solutions of the sub-Cauchy–Stokes problem using matching conditions defined on the inaccessible boundary. The matching is based on second order conditions and the types of the interfacial equations depend on the equations used to match the values of the unknowns on the inaccessible boundary. The interfacial problems are then solved by iterative procedures in which coefficients can be optimized to improve convergence rates. A complete analysis of the methods is presented, and intensive numerical results illustrate the effectiveness and the performance of the proposed approaches.

Journal

Journal of Computational and Applied MathematicsElsevier

Published: Aug 15, 2018

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches

$49/month

Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.

$588

$360/year

billed annually
Start Free Trial

14-day Free Trial