The combination technique for a two-dimensional convection-diffusion problem with exponential layers

Sebastian Franz; Fang Liu; Hans-Görg Roos; Martin Stynes; Aihui Zhou

Applications of Mathematics (2009)

  • Volume: 54, Issue: 3, page 203-223
  • ISSN: 0862-7940

Abstract

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Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a sparse grid Galerkin finite element method with Shishkin meshes. Writing N for the maximum number of mesh intervals in each coordinate direction, our “combination” method simply adds or subtracts solutions that have been computed by the Galerkin FEM on N × N , N × N and N × N meshes. It is shown that the combination FEM yields (up to a factor ln N ) the same order of accuracy in the associated energy norm as the Galerkin FEM on an N × N mesh, but it requires only 𝒪 ( N 3 / 2 ) degrees of freedom compared with the 𝒪 ( N 2 ) used by the Galerkin FEM. An analogous result is also proved for the streamline diffusion finite element method.

How to cite

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Franz, Sebastian, et al. "The combination technique for a two-dimensional convection-diffusion problem with exponential layers." Applications of Mathematics 54.3 (2009): 203-223. <http://eudml.org/doc/37816>.

@article{Franz2009,
abstract = {Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a sparse grid Galerkin finite element method with Shishkin meshes. Writing $N$ for the maximum number of mesh intervals in each coordinate direction, our “combination” method simply adds or subtracts solutions that have been computed by the Galerkin FEM on $N \times \sqrt\{N\}$, $\sqrt\{N\} \times N$ and $\sqrt\{N\} \times \sqrt\{N\}$ meshes. It is shown that the combination FEM yields (up to a factor $\ln N$) the same order of accuracy in the associated energy norm as the Galerkin FEM on an $N\times N$ mesh, but it requires only $\mathcal \{O\}(N^\{3/2\})$ degrees of freedom compared with the $\mathcal \{O\}(N^2)$ used by the Galerkin FEM. An analogous result is also proved for the streamline diffusion finite element method.},
author = {Franz, Sebastian, Liu, Fang, Roos, Hans-Görg, Stynes, Martin, Zhou, Aihui},
journal = {Applications of Mathematics},
keywords = {convection-diffusion; finite element; Shishkin mesh; two-scale discretization; exponential layers; Galerkin FEM; convection-diffusion; finite element; Shishkin mesh; two-scale discretization; exponential layers; Galerkin FEM},
language = {eng},
number = {3},
pages = {203-223},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The combination technique for a two-dimensional convection-diffusion problem with exponential layers},
url = {http://eudml.org/doc/37816},
volume = {54},
year = {2009},
}

TY - JOUR
AU - Franz, Sebastian
AU - Liu, Fang
AU - Roos, Hans-Görg
AU - Stynes, Martin
AU - Zhou, Aihui
TI - The combination technique for a two-dimensional convection-diffusion problem with exponential layers
JO - Applications of Mathematics
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 3
SP - 203
EP - 223
AB - Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a sparse grid Galerkin finite element method with Shishkin meshes. Writing $N$ for the maximum number of mesh intervals in each coordinate direction, our “combination” method simply adds or subtracts solutions that have been computed by the Galerkin FEM on $N \times \sqrt{N}$, $\sqrt{N} \times N$ and $\sqrt{N} \times \sqrt{N}$ meshes. It is shown that the combination FEM yields (up to a factor $\ln N$) the same order of accuracy in the associated energy norm as the Galerkin FEM on an $N\times N$ mesh, but it requires only $\mathcal {O}(N^{3/2})$ degrees of freedom compared with the $\mathcal {O}(N^2)$ used by the Galerkin FEM. An analogous result is also proved for the streamline diffusion finite element method.
LA - eng
KW - convection-diffusion; finite element; Shishkin mesh; two-scale discretization; exponential layers; Galerkin FEM; convection-diffusion; finite element; Shishkin mesh; two-scale discretization; exponential layers; Galerkin FEM
UR - http://eudml.org/doc/37816
ER -

References

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