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The combination technique for a two-dimensional convection-diffusion problem with exponential layers. (English) Zbl 1212.65443
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 (FEM) 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.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
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