AbstractWe 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. In the case of rectangular meshes
which are (a priori or a posteriori ) locally condensed in the transition layer. 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’.
A linearised singularly perturbed convection–diffusion problem with an interior layer
