Convection dominated singularly perturbed problems on a metric graph

Abstract

This article concentrates on singularly perturbed static convection–diffusion equations with varying coefficients on a metric graph G=(V,E). Our interest is in the convection dominated situation which is described by a small parameter ϵ>0 in front of the diffusion term. As ϵ→0, the reduced problem may exhibit boundary layers at the multiple vertices as well as at the simple nodes. We analyze the possible scenarios and validate the results in several test cases. We investigate several exemplary graphs and use an upwind finite difference method on a piece-wise Shishkin mesh. Error estimates are also discussed to show ϵ-uniform convergence.