Numerical convergence of a parameterisation method for the solution of a highly anisotropic two-dimensional elliptic problem.
Résumé
Highly anisotropic two-dimensional elliptic problems lead to severe numerical difficulties. In this paper, starting from a simple finite volume scheme, we present a parameterisation method that allows us to obtain the solution even if the anisotropy ratio is very large. We derive a formal asymptotic limit of the two dimensional anisotropic problem, in the case where the anisotropy ratio goes to infinity. This formal limit is used as a reference, and we show that the parameterisation method gives similar results, whereas the finite volume scheme fails to give an accurate solution. Numerical results are given, which indicate important parameters to be considered in order to obtain a good precision.