Numerical resolution of an anisotropic non-linear diffusion problem

This paper is devoted to the numerical resolution of an anisotropic non-linear diffusion problem involving a small parameter ǫ, defined as the anisotropy strength reciprocal. In this work, the anisotropy is carried by a variable vector function b. The equation being supplemented with Neumann boundary conditions, the limit ǫ→0 is demonstrated to be a singular perturbation of the original diffusion equation. To address efficiently this problem, an Asymptotic-Preserving scheme is derived. This numerical method does not require the use of coordinates adapted to the anisotropy direction and exhibits an accuracy as well as a computational cost independent of the anisotropy strength. keywords Anisotropic diffusion problems; Singular perturbation; Asymptotic-Preserving schemes. AMS Subject Classification 35J60, 35J62, 65M06, 65M12, 65N06, 65N12.