Commun. Comput. Phys., 9 (2011), pp. 1020-1039.

A Comparison of Semi-Lagrangian and Lagrange-Galerkin hp-FEM Methods in Convection-Diffusion Problems

Pedro Galan del Sastre 1*, Rodolfo Bermejo 2

1 Departamento de Matematica Aplicada al Urbanismo, a la Edificacion y al Medio Ambiente, E.T.S.A.M., Universidad Politecnica de Madrid, Avda. Juan de Herrera 4, 28040 Madrid, Spain.
2 Departamento de Matematica Aplicada, E.T.S.I.I., Universidad Politecnica de Madrid, C/ Jose Gutierrez Abascal 2, 28006 Madrid, Spain.

Received 4 December 2009; Accepted (in revised version) 16 September 2010
Available online 10 November 2010


We perform a comparison in terms of accuracy and CPU time between second order BDF semi-Lagrangian and Lagrange-Galerkin schemes in combination with high order finite element method. The numerical results show that for polynomials of degree 2 semi-Lagrangian schemes are faster than Lagrange-Galerkin schemes for the same number of degrees of freedom, however, for the same level of accuracy both methods are about the same in terms of CPU time. For polynomials of degree larger than 2, Lagrange-Galerkin schemes behave better than semi-Lagrangian schemes in terms of both accuracy and CPU time; specially, for polynomials of degree 8 or larger. Also, we have performed tests on the parallelization of these schemes and the speed-up obtained is quasi-optimal even with more than 100 processors.

AMS subject classifications: 65M60, 65L60, 65N30, 65M25, 76M10, 76M25

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Key words: Navier-Stokes equations, convection-diffusion equations, semi-Lagrangian, Lagrange-Galerkin, second order backward difference formula, hp-finite element method.

*Corresponding author.
Email: (P. Galan del Sastre), (R. Bermejo)

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