Catalcg ing-in-Publ ication Data applied forDie Deutsche Bibliothek -CIP-Einheitsaufnahme Roos, Hans-Görg: Nume rica\ meth ods for singularly perturbed diff erential equat io ns : con vecti on diffus ion and flow pr obl ems / H .-G.
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.
Error estimates of finite element methods for reaction-diffusion problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the H 1 seminorm leads to a balanced norm which reflects the layer behavior correctly. We prove error estimates in balanced norms and investigate also stability questions. Especially, we propose a new C 0 interior penalty method with improved stability properties in comparison with the Galerkin FEM.
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