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Limitations of adaptive mesh re$nement techniques for singularly perturbed problems with a moving interior layer
| Content Provider | Semantic Scholar |
|---|---|
| Author | Shishkina, G. I. |
| Copyright Year | 2004 |
| Abstract | In a composed domain on an axis R with the moving interface boundary between two subdomains, we consider an initial value problem for a singularly perturbed parabolic reaction–di3usion equation in the presence of a concentrated source on the interface boundary. Monotone classical di3erence schemes for problems from this class converge only when N−1 + N−1 0 , where is the perturbation parameter, N and N0 de$ne the number of mesh points with respect to x (on segments of unit length) and t. Therefore, in the case of such problems with moving interior layers, it is necessary to develop special numerical methods whose errors depend rather weakly on the parameter and, in particular, are independent of (i.e., -uniformly convergent methods). In this paper we study schemes on adaptive meshes which are locally condensing in a neighbourhood of the set ∗, that is, the trajectory of the moving source. It turns out, that in the class of di3erence schemes consisting of a standard $nite di3erence operator on rectangular meshes which are (a priori or a posteriori) locally condensing in x and t, there are no schemes that converge -uniformly, and in particular, even under the condition ≈ N−2 +N−2 0 , if the total number of the mesh points between the cross-sections x0 and x0 + 1 for any x0 ∈R has order of NN0. Thus, the adaptive mesh re$nement techniques used directly do not allow us to widen essentially the convergence range of classical numerical methods. On the other hand, the use of This work was supported partially by the Russian Foundation for Basic Research under Grant No. 01–01–01022 and by the Enterprise Ireland Grant SC–2000–070. ∗ Address for correspondence: Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, 16 S. Kovalevskaya Street, Ekaterinburg 620219, Russia. E-mail addresses: shishkin@imm.uran.ru, shishkin@maths.tcd.ie (G.I. Shishkin). 0377-0427/$ see front matter c © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2003.09.022 268 G.I. Shishkin / Journal of Computational and Applied Mathematics 166 (2004) 267–280 condensing meshes but in a local coordinate system $tted to the set ∗ makes it possible to construct schemes which converge -uniformly for N; N0 → ∞; such a scheme converges at the rate O(N−1 ln N + N−1 0 ). c © 2003 Elsevier B.V. All rights reserved. MSC: 65M06; 65M15; 65M50; 35B25 |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://core.ac.uk/download/pdf/81187218.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
