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Volume 11, Issue 2
Pure Lagrangian and Semi-Lagrangian Finite Element Methods for the Numerical Solution of Convection-Diffusion Problems

M. Benitez & A. Bermudez

Int. J. Numer. Anal. Mod., 11 (2014), pp. 271-287.

Published online: 2014-11

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  • Abstract

In this paper we propose a unified formulation to introduce and analyze (pure) Lagrangian and semi-Lagrangian methods for solving convection-diffusion partial differential equations. This formulation allows us to state classical and new numerical methods. Several examples are given. We combine them with finite element methods for spatial discretization. One of the pure Lagrangian methods we introduce has been analyzed in [4] and [5] where stability and error estimates for time semi-discretized and fully-discretized schemes have been proved. In this paper, we prove new stability estimates. More precisely, we obtain an $l^∞(H^1)$ stability estimate independent of the diffusion coefficient and, if the underlying flow is incompressible, we get a stability inequality independent of the final time. Finally, the numerical solution of a test problem is presented that confirms the new stability results.

  • Keywords

convection-diffusion equation, pure Lagrangian method, semi-Lagrangian method, Lagrange-Galerkin method, characteristics method, second order schemes, finite element method.

  • AMS Subject Headings

35R35, 49J40, 60G40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-11-271, author = {M. Benitez and A. Bermudez}, title = {Pure Lagrangian and Semi-Lagrangian Finite Element Methods for the Numerical Solution of Convection-Diffusion Problems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2014}, volume = {11}, number = {2}, pages = {271--287}, abstract = {

In this paper we propose a unified formulation to introduce and analyze (pure) Lagrangian and semi-Lagrangian methods for solving convection-diffusion partial differential equations. This formulation allows us to state classical and new numerical methods. Several examples are given. We combine them with finite element methods for spatial discretization. One of the pure Lagrangian methods we introduce has been analyzed in [4] and [5] where stability and error estimates for time semi-discretized and fully-discretized schemes have been proved. In this paper, we prove new stability estimates. More precisely, we obtain an $l^∞(H^1)$ stability estimate independent of the diffusion coefficient and, if the underlying flow is incompressible, we get a stability inequality independent of the final time. Finally, the numerical solution of a test problem is presented that confirms the new stability results.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/525.html} }
TY - JOUR T1 - Pure Lagrangian and Semi-Lagrangian Finite Element Methods for the Numerical Solution of Convection-Diffusion Problems AU - M. Benitez & A. Bermudez JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 271 EP - 287 PY - 2014 DA - 2014/11 SN - 11 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/525.html KW - convection-diffusion equation, pure Lagrangian method, semi-Lagrangian method, Lagrange-Galerkin method, characteristics method, second order schemes, finite element method. AB -

In this paper we propose a unified formulation to introduce and analyze (pure) Lagrangian and semi-Lagrangian methods for solving convection-diffusion partial differential equations. This formulation allows us to state classical and new numerical methods. Several examples are given. We combine them with finite element methods for spatial discretization. One of the pure Lagrangian methods we introduce has been analyzed in [4] and [5] where stability and error estimates for time semi-discretized and fully-discretized schemes have been proved. In this paper, we prove new stability estimates. More precisely, we obtain an $l^∞(H^1)$ stability estimate independent of the diffusion coefficient and, if the underlying flow is incompressible, we get a stability inequality independent of the final time. Finally, the numerical solution of a test problem is presented that confirms the new stability results.

M. Benitez and A. Bermudez. (2014). Pure Lagrangian and Semi-Lagrangian Finite Element Methods for the Numerical Solution of Convection-Diffusion Problems. International Journal of Numerical Analysis and Modeling. 11 (2). 271-287. doi:
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