Opuscula Math. 37, no. 2 (2017), 327-345
http://dx.doi.org/10.7494/OpMath.2017.37.2.327

Opuscula Mathematica

# The interaction between PDE and graphs in multiscale modeling

Fernando A. Morales
Sebastián Naranjo Álvarez

Abstract. In this article an upscaling model is presented for complex networks with highly clustered regions exchanging/trading quantities of interest at both, microscale and macroscale level. Such an intricate system is approximated by a partitioned open map in $$\mathbb{R}^{2}$$ or $$\mathbb{R}^{3}$$. The behavior of the quantities is modeled as flowing in the map constructed and thus it is subject to be described using partial differential equations. We follow this approach using the Darcy Porous Media, saturated fluid flow model in mixed variational formulation.

Keywords: coupled PDE systems, mixed formulations, porous media, analytic graph theory, complex networks.

Mathematics Subject Classification: 05C82, 05C10, 35R02, 35J50.

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1. G. Berkolaiko, P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, American Mathematical Society, Rhode Island, 2012.
2. A. Bondy, U. Murty, Graph Theory, Graduate Texts in Mathematics, vol. 244, Springer, 2008.
3. F.-R.K. Chung, Spectral Graph Theory, Regional Conference Series in Mathematics, vol. 92, American Mathematical Society, Rhode Island, 1994.
4. S.-Y. Chung, Y.-S. Chung, J.-H. Kim, Diffusion and elastic equations on networks, Publ. RIMS 43 (2007), 699-726.
5. E. Estrada, The Structure of Complex Networks, Theory and Applications, Oxford University Press, New York, 2012.
6. J. Friedman, J.-P. Tillich, Wave equations for graphs and the edge-based Laplacian, Pacific J. Math. 216 (2004), 699-266.
7. V. Girault, P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, vol. 749 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1979.
8. G.R.C. Godsil, Algebraic Graph Theory, Graduate Texts in Mathematics, Springer, New York, 2001.
9. M.O. Jackson, Social and Economic Networks, Princeton University Press, New Jersey, 2008.
10. F. Morales, R. Showalter, Interface approximation of Darcy flow in a narrow channel, Math. Methods Appl. Sci. 35 (2012), 182-195.
11. F.A. Morales, Homogenization of geological fissured systems with curved non-periodic cracks, Electron. J. Differential Equations 2014 (2014) 189, 1-29.
12. F.A. Morales, M.A. Osorio, On the generation of bipartite grids with controlled regularity for 2-d and 3-d simply connected domains, Appl. Anal. Discrete Math. 8 (2014) 173-199.
13. L. Orsenigo, Clusters and clustering in biotechnology: Stylised facts, issues and theories, [in:] P. Braunerhjelm, M.P. Feldman (eds), Cluster Genesis, Oxford University Press, 2006, 195-218.
14. J. Solomon, PDE approaches to graph analysis, arXiv:1505.00185[cs.DM], 2015.
15. M. van Steen, Graph Theory and Complex Networks, an Introduction, Maarten van Steen, Amsterdam, 2010.
16. R.J. Wilson, Introduction to Graph Theory, Addison Wesley Longman Limited, Essex CM20 2JE, England, 1996.
• Fernando A. Morales
• Universidad Nacional de Colombia, Escuela de Matemáticas, Sede Medellín, Colombia
• Sebastián Naranjo Álvarez
• Oregon State University, Department of Mathematics, Corvallis, OR, 97331-4605, USA
• Communicated by P.A. Cojuhari.
• Accepted: 2016-07-25.
• Published online: 2017-01-03.