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.

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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 $N \times \sqrt N$, $\sqrt N \times N$ 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 
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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. http://geodesic.mathdoc.fr/articles/10.1007/s10492-009-0013-9/

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