| Issue |
ESAIM: M2AN
Volume 57, Number 4, July-August 2023
|
|
|---|---|---|
| Page(s) | 2077 - 2095 | |
| DOI | https://doi.org/10.1051/m2an/2023044 | |
| Published online | 03 July 2023 | |
A hybrid-dG method for singularly perturbed convection-diffusion equations on pipe networks
1
Institute for Numerical Mathematics, Johannes-Kepler University Linz, Linz, Austria
2
Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria
* Corresponding author: nora.philippi@ricam.oeaw.ac.at
Received:
9
September
2022
Accepted:
16
May
2023
We study the numerical approximation of singularly perturbed convection-diffusion problems on one-dimensional pipe networks. In the vanishing diffusion limit, the number and type of boundary conditions and coupling conditions at network junctions change, which gives rise to singular layers at the outflow boundaries of the pipes. A hybrid discontinuous Galerkin method is proposed, which provides a natural upwind mechanism for the convection-dominated case. Moreover, the method provides a viable approximation for the limiting pure transport problem. A detailed analysis of the singularities of the solution and the discretization error is presented, and an adaptive strategy is proposed, leading to order optimal error estimates that hold uniformly in the singular perturbation limit. The theoretical results are confirmed by numerical tests.
Mathematics Subject Classification: 35B25 / 35B40 / 35K20 / 35R02 / 65N30 / 76R99
Key words: Singular perturbation problems / vanishing diffusion limit / discontinuous Galerkin methods / asymptotic analysis / parameter robust error estimates
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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