Geodesic
The combination technique for a two-dimensional convection-diffusion problem with exponential layers
Applications of Mathematics, Tome 54 (2009) no. 3, pp. 203-223
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $\Cal O(N^{3/2})$ degrees of freedom compared with the $\Cal O(N^2)$ used by the Galerkin FEM. An analogous result is also proved for the streamline diffusion finite element method.
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 $\Cal O(N^{3/2})$ degrees of freedom compared with the $\Cal O(N^2)$ used by the Galerkin FEM. An analogous result is also proved for the streamline diffusion finite element method.
DOI : 10.1007/s10492-009-0013-9
Classification : 65F10, 65N15, 65N30, 65N55, 65Y10
Keywords: convection-diffusion; finite element; Shishkin mesh; two-scale discretization; exponential layers; Galerkin FEM 
@article{10_1007_s10492_009_0013_9,
     author = {Franz, Sebastian and Liu, Fang and Roos, Hans-G\"org and Stynes, Martin and Zhou, Aihui},
     title = {The combination technique for a two-dimensional convection-diffusion problem with exponential layers},
     journal = {Applications of Mathematics},
     pages = {203--223},
     year = {2009},
     volume = {54},
     number = {3},
     doi = {10.1007/s10492-009-0013-9},
     mrnumber = {2530539},
     zbl = {1212.65443},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10492-009-0013-9/}
}
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
SP  - 203
EP  - 223
VL  - 54
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10492-009-0013-9/
DO  - 10.1007/s10492-009-0013-9
LA  - en
ID  - 10_1007_s10492_009_0013_9
ER  - 
%0 Journal Article
%A Franz, Sebastian
%A Liu, Fang
%A Roos, Hans-Görg
%A Stynes, Martin
%A Zhou, Aihui
%T The combination technique for a two-dimensional convection-diffusion problem with exponential layers
%J Applications of Mathematics
%D 2009
%P 203-223
%V 54
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1007/s10492-009-0013-9/
%R 10.1007/s10492-009-0013-9
%G en
%F 10_1007_s10492_009_0013_9
Franz, Sebastian; Liu, Fang; Roos, Hans-Görg; Stynes, Martin; Zhou, Aihui. The combination technique for a two-dimensional convection-diffusion problem with exponential layers. Applications of Mathematics, Tome 54 (2009) no. 3, pp. 203-223. doi: 10.1007/s10492-009-0013-9

Cité par Sources :