A Class of Singularly Perturbed Convection-diffusion Problems with a Moving Interior Layer. an a Posteriori Adaptive Mesh Technique

We study numerical approximations for a class of singularly perturbed convection-diffusion type problems with a moving interior layer. In a domain (segment) with a moving interface between two subdomains, we consider an initial boundary value problem for a singularly perturbed parabolic convection-diffusion equation. Convection fluxes on the subdomains are directed towards the interface. The solution of this problem has a moving transition layer in the neighbourhood of the interface. Unlike problems with a stationary layer, the solution exhibits singular behaviour also with respect to the time variable. Well-known upwind finite difference schemes for such problems do not converge ε-uniformly in the uniform norm, even under the condition N−1 + N−1 0 ≈ ε, where ε is the perturbation parameter and N and N0 denote the number of mesh points with respect to x and t. In the case of rectangular meshes which are (a priori or a posteriori ) locally condensed in the transition layer, there are no schemes that converge uniformly in ε even under the very restrictive condition N−2 +N−2 0 ≈ ε. However, the condition for convergence can be considerably weakened if we take the geometry of the layer into account, i.e., if we introduce a new coordinate system which captures the interface. For the problem in such a coordinate system, one can use either an a priori, or an a posteriori adaptive mesh technique. Here we construct a scheme on a posteriori adaptive meshes (based on the solution gradient), whose solution converges ‘almost ε-uniformly’, viz., under the condition N−1 = o(ε), where ν > 0 is an arbitrary number from the half-open interval (0, 1]. 2000 Mathematics Subject Classification: 65M06; 65M12; 65M15.