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A Discontinuous Galerkin and Semismooth Newton Approach for the Numerical Solution of Bingham Flow with Variable Density

  • Sergio González-Andrade ORCID logo EMAIL logo and Paul E. Méndez Silva ORCID logo
Published/Copyright: August 25, 2023

Abstract

This paper is devoted to the study of Bingham flow with variable density. We propose a local bi-viscosity regularization of the stress tensor based on a Huber smoothing step. Next, our computational approach is based on a second-order, divergence-conforming discretization of the Huber regularized Bingham constitutive equations, coupled with a discontinuous Galerkin scheme for the mass density. We take advantage of the properties of divergence-conforming and discontinuous Galerkin formulations to effectively incorporate upwind discretizations, thereby ensuring the stability of the formulation. The stability of the continuous problem and the fully discrete scheme are analyzed. Further, a semismooth Newton method is proposed for solving the obtained fully discretized system of equations at each time step. Finally, several numerical examples that illustrate the main features of the problem and the properties of the numerical scheme are presented.

MSC 2010: 76A05; 76-10; 76M10; 65M60; 49M15

Award Identifier / Grant number: PIS 18-03

Award Identifier / Grant number: PIGR 19-02

Funding statement: We acknowledge the partial support by Escuela Politécnica Nacional del Ecuador, under the projects PIS 18-03 and PIGR 19-02.

Acknowledgements

We are grateful to the anonymous reviewers whose comments helped us to improve the article. This research was carried out by using the research computing facilities offered by the Scientific Computing Laboratory of the Research Center on Mathematical Modeling: MODEMAT, Escuela Politécnica Nacional – Quito.

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Received: 2022-11-15
Revised: 2023-07-11
Accepted: 2023-08-03
Published Online: 2023-08-25
Published in Print: 2024-04-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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