Pure Lagrangian and Semi-Lagrangian Finite Element Methods for the Numerical Solution of Convection-Diffusion Problems
Year: 2014
International Journal of Numerical Analysis and Modeling, Vol. 11 (2014), Iss. 2 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2014-IJNAM-525
International Journal of Numerical Analysis and Modeling, Vol. 11 (2014), Iss. 2 : pp. 271–287
Published online: 2014-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 17
Keywords: convection-diffusion equation pure Lagrangian method semi-Lagrangian method Lagrange-Galerkin method characteristics method second order schemes finite element method.