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Article

Franz, Sebastian ; Liu, Fang ; Roos, Hans-Görg ; Stynes, Martin ; Zhou, Aihui
The combination technique for a two-dimensional convection-diffusion problem with exponential layers. (English). Applications of Mathematics, vol. 54 (2009), issue 3, pp. 203-223
MSC: 65F10, 65N15, 65N30, 65N55, 65Y10 | MR 2530539 | Zbl 1212.65443 | DOI: 10.1007/s10492-009-0013-9
Keywords:
convection-diffusion; finite element; Shishkin mesh; two-scale discretization; exponential layers; Galerkin FEM
Summary:
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.
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