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Journal of Scientific Computing

, Volume 65, Issue 3, pp 1270–1308 | Cite as

Analysis of an Augmented HDG Method for a Class of Quasi-Newtonian Stokes Flows

  • Gabriel N. Gatica
  • Filánder A. Sequeira
Article

Abstract

In this paper we introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for numerically solving a class of nonlinear Stokes models arising in quasi-Newtonian fluids. Similarly as in previous papers dealing with the application of mixed finite element methods to these nonlinear models, we use the incompressibility condition to eliminate the pressure, and set the velocity gradient as an auxiliary unknown. In addition, we enrich the HDG formulation with two suitable augmented equations, which allows us to apply known results from nonlinear functional analysis, namely a nonlinear version of Babuška–Brezzi theory and the classical Banach fixed-point theorem, to prove that the discrete scheme is well-posed and derive the corresponding a priori error estimates. Then we discuss some general aspects concerning the computational implementation of the method, which show a significant reduction of the size of the linear systems involved in the Newton iterations. Finally, we provide several numerical results illustrating the good performance of the proposed scheme and confirming the optimal order of convergence provided by the HDG approximation.

Keywords

Nonlinear Stokes model Mixed finite element method  Hybridized discontinuous Galerkin method Augmented formulation 

Notes

Acknowledgments

The authors are thankful to Paul Castillo and Manuel Solano for valuable remarks concerning the computational implementation of the HDG method.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Centro de Investigación en Ingeniería Matemática, Departamento de Ingeniería MatemáticaUniversidad de ConcepciónConcepciónChile
  2. 2.Escuela de MatemáticaUniversidad Nacional de Costa RicaHerediaCosta Rica
  3. 3.Centro de Investigación en Ingeniería Matemática, Departamento de Ingeniería MatemáticaUniversidad de ConcepciónConcepciónChile

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