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Thus, the discrete solution oscillates mildly and the oscillations decay to zero as N increases. Therefore we conclude:A thin submesh leads to some stabilization in the upstream direction. Fig. 1 shows the numerical solution for the two-dimensional problem À10 À3 Du þ ð2 À xÞ u x þ ð3 À yÞ u y þ 2u ¼ xð1 À xÞ with homogeneous boundary conditions on the unit square based on central differencing and a tensor-product mesh of the type just described with a small h 2 as for a 24Â24 and 48Â48 Shishkin mesh (from the fine submesh just one black line is visible). While the solution on a uniform mesh is characterized by wild oscillations here the fine submesh improves the situation upwind from the fine submesh a lot. References1 Andreev, V. B.; Kopteva, N. V.: On the investigation of difference schemes with a central approximation of the first order derivative. Zh. Vychisl. Mat. Mat. Fiz. 36 (1996), 101--117 (Russian). 2 Lenferink, H. W. J.: Some superconvergence results for finite element discretizations on a Shishkin mesh of a convection-diffusion problem.In this study, the determination of meso-shape functions for the meso-structural model of a wavy-plates is investigated.

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We consider a singularly perturbed elliptic convection᎐diffusion problem on the unit square. A new asymptotic expansion of its solution is constructed, giving precise conditions under which the solution can be decomposed in a particularly opportune way into a sum of smooth and layer functions.

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We consider singularly perturbed linear elliptic problems in two dimensions. The solutions of such problems typically exhibit layers and are difficult to solve numerically. The streamline diffusion finite element method (SDFEM) has been proved to produce accurate solutions away from any layers on uniform meshes, but fails to compute the boundary layers precisely. Our modified SDFEM is implemented with piecewise linear functions on a Shishkin mesh that resolves boundary layers, and we prove that it yields an accurate approximation of the solution both inside and outside these layers. The analysis is complicated by the severe nonuniformity of the mesh. We give local L ∞ error estimates that hold true uniformly in the perturbation parameter ε, provided only that ε ≤ N −1 , where O(N 2 ) mesh points are used. Numerical experiments support these theoretical results.Mathematics Subject Classification (1991): 65N06, 65N15, 65N50