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On the transport limit of singularly perturbed convection–diffusion problems on networks
Author(s) -
Egger Herbert,
Philippi Nora
Publication year - 2021
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.7084
Subject(s) - mathematics , limit (mathematics) , convection–diffusion equation , dissipation , boundary value problem , norm (philosophy) , diffusion , singular perturbation , mathematical analysis , convection , asymptotic analysis , coupling (piping) , physics , mechanics , mechanical engineering , political science , law , thermodynamics , engineering
We consider singularly perturbed convection–diffusion equations on one‐dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling conditions at inner vertices are derived that guarantee conservation of mass and dissipation of a mathematical energy which allows us to prove stability and well‐posedness. For single intervals and appropriately specified initial conditions, it is well‐known that the solutions of the convection–diffusion problem converge to that of the transport problem with order O ( ϵ ) in the L ∞ ( L 2 ) ‐norm with diffusion ϵ → 0 . In this paper, we prove a corresponding result for problems on one‐dimensional networks. The main difficulty in the analysis is that the number and type of coupling conditions changes in the singular limit which gives rise to additional boundary layers at the interior vertices of the network. Since the values of the solution at these network junctions are not known a priori, the asymptotic analysis requires a delicate choice of boundary layer functions that allows to handle these interior layers.