Limitations of Adaptive Mesh Refinement Methods for a Singularly Perturbed Parabolic Equation in a Composed Domain with a Concentrated Source on the Moving Interface Boundary

In a composed domain on an axis $I\!\!R\,$ 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 $\eps \gg N^{-1}+N_0^{-1}$, where $\eps$ 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 $\eps$ and, in particular, are independent~of~$\eps$ (i.e. $\eps$-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 ({\it a~priori\,} or {\it a~posteriori\,}) locally condensing~in~$x$~and~$t$ there are no schemes which converge $\eps$-uniformly and, in particular, even under the condition $\eps\!\approx N^{-2}+N_0^{-2}\,$ if the total number of the mesh points between the cross-sections $x_0$ and $x_0+1$ for any $x_0\in I\!\!R\,$ has order of $NN_0$. 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 $\eps$-uniformly for $N,\, N_0 \to \infty$; such a scheme converges with the rate~$\Oh{N^{-1} \ln N + N_0^{-1}}$.