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

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 ε⪢N−1+N0−1, where ε is the perturbation parameter, N and N0 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 ε and, in particular, are independent of ε (i.e., ε-uniformly convergent methods).In this paper we study schemes on adaptive meshes which are locally condensing in a neighbourhood of the set γ∗, 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 ε-uniformly, and in particular, even under the condition ε≈N−2+N0−2, 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 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 γ∗ makes it possible to construct schemes which converge ε-uniformly for N,N0→∞; such a scheme converges at the rate O(N−1lnN+N0−1).