Strongly anisotropic diffusion problems; asymptotic analysis

TL;DR

This paper investigates anisotropic diffusion equations with highly directional diffusion matrices, providing asymptotic approximations, analyzing well-posedness, and establishing convergence using averaging techniques.

Contribution

It introduces first and second order asymptotic approximations for anisotropic diffusion equations with disparate eigenvalues and analyzes their well-posedness and convergence.

Findings

Derived first and second order asymptotic approximations.

Proved well-posedness of the approximations.

Established convergence results for the asymptotic analysis.

Abstract

The subject matter of this paper concerns anisotropic diffusion equations: we consider heat equations whose diffusion matrix have disparate eigenvalues. We determine first and second order approximations, we study the well-posedness of them and establish convergence results. The analysis relies on averaging techniques, which have been used previously for studying transport equations whose advection fields have disparate components.