<abstract xmlns="http://www.w3.org/1999/xhtml"><p> We review in this paper the development of Lagrange-Galerkin (LG) methods to integrate the incompressible Navier-Stokes equations (NSEs) for engineering applications. These methods were introduced in the computational fluid dynamics community in the early eighties of the past century, and at that time they were considered good methods for both their theoretical stability properties and the way of dealing with the nonlinear terms of the equations; however, the numerical experience gained with the application of LG methods to different problems has identified drawbacks of them, such as the calculation of specific integrals that arise in their formulation and the calculation of the ow trajectories, which somehow have hampered the applicability of LG methods. In this paper, we focus on these issues and summarize the convergence results of LG methods; furthermore, we shall briefly introduce a new stabilized LG method suitable for high Reynolds numbers.</p></abstract>

We review in this paper the development of Lagrange-Galerkin (LG) methods to integrate the incompressible Navier-Stokes equations (NSEs) for engineering applications. These methods were introduced in the computational fluid dynamics community in the early eighties of the past century, and at that time they were considered good methods for both their theoretical stability properties and the way of dealing with the nonlinear terms of the equations; however, the numerical experience gained with the application of LG methods to different problems has identified drawbacks of them, such as the calculation of specific integrals that arise in their formulation and the calculation of the ow trajectories, which somehow have hampered the applicability of LG methods. In this paper, we focus on these issues and summarize the convergence results of LG methods; furthermore, we shall briefly introduce a new stabilized LG method suitable for high Reynolds numbers.

Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review

Departamento de Matemática Aplicada a la Ingeniería Industrial ETSII , Universidad Politecnica de Madrid,

Departamento de Matemática Aplicada a la Ingeniería Aeroespacial ETSAA, Universidad Politecnica de Madrid,

Communications in Applied and Industrial Mathematics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

We review in this paper the development of Lagrange-Galerkin (LG) methods to integrate the incompressible Navier-Stokes equations (NSEs) for engineering...

Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review

Published Online: Oct 01, 2016

Page range: 26 - 55

Received: May 10, 2015

Accepted: Nov 09, 2015

DOI: https://doi.org/10.1515/caim-2016-0021

Keywords
Lagrange-Galerkin, finite elements, Navier-Stokes

© 2016

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review

Published Online: Oct 01, 2016

Page range: 26 - 55

Received: May 10, 2015

Accepted: Nov 09, 2015

DOI: https://doi.org/10.1515/caim-2016-0021

KeywordsLagrange-Galerkin, finite elements, Navier-Stokes

© 2016

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Lagrange-Galerkin, finite elements, Navier-Stokes