Limitations of adaptive mesh refinement techniques for singularly perturbed problems with a moving interior layer

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:166

Article; Proceedings Paper

Shishkin, GI

关键词: singularly perturbed parabolic equation;    moving interior layer;    finite difference methods;    epsilon-uniform convergence;    adaptive mesh refinement;    Kolmogorov widths;   

DOI  :  10.1016/j.cam.2003.09.022

来源: Elsevier

【 摘 要 】

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-diffusion equation in the presence of a concentrated source on the interface boundary. Monotone classical difference schemes for problems from this class converge only when epsilonmuch greater thanN(-1) + N-0(-1), where epsilon is the perturbation parameter, N and N-0 define 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 epsilon and, in particular, are independent of epsilon (i.e., epsilon-uniformly convergent methods). In this paper we study schemes on adaptive meshes which are locally condensing in a neighbourhood of the set gamma*, that is, the trajectory of the moving source. It turns out, that in the class of difference schemes consisting of a standard finite difference operator on rectangular meshes which are (a priori or a posteriori) locally condensing in x and t, there are no schemes that converge epsilon-uniformly, and in particular, even under the condition epsilon approximate to N-2 + N-0(-2), if the total number of the mesh points between the cross-sections x(0) and x(0) + 1 for any x(0) is an element ofR has order of NN0. Thus, the adaptive mesh refinement techniques used directly do not allow us to widen essentially the convergence range of classical numerical methods. On the other hand, the use of condensing meshes but in a local coordinate system fitted to the set gamma* makes it possible to construct schemes which converge epsilon-uniformly for N, N-0 --> infinity; such a scheme converges at the rate O(N-1 ln N + N-0(-1)). (C) 2003 Elsevier B.V. All rights reserved.

【 授权许可】

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