Notes on the nonlinear dependence of a multiscale coupled system with respect to the interface

• Autorzy: Morales F. A.

Identyfikatory

DOI

http://dx.doi.Org/10.7494/OpMath.2015.35.4.517

• Języki publikacji: EN

Abstrakty

EN

This work studies the dependence of the solution with respect to interface geometric perturbations, in a multiscaled coupled Darcy flow system in direct variational formulation. A set of admissible perturbation functions and a sense of convergence is presented, as well as sufficient conditions on the forcing terms, in order to conclude strong convergence statements. For the rate of convergence of the solutions we start solving completely the one dimensional case, using orthogonal decompositions on the appropriate subspaces. Finally, the rate of convergence question is analyzed in a simple multi dimensional setting, by studying the nonlinear operators introduced due to the geometric perturbations.

Słowa kluczowe

EN

multiscale coupled systems; interface geometric perturbations; variational formulations; nonlinear dependence

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2015
• Tom: Vol. 35, no. 4
• Strony: 517--546
• Opis fizyczny: Bibliogr. 23 poz., rys.

Twórcy

autor

Morales F. A.

• Universidad Nacional de Colombia Escuela de Matematicas Sede Medellin Calle 59 A No 63-20, Offlcina 43-106 Medellin, Colombia

Bibliografia

• [1] T. Arbogast, H. Lehr, Homogenization of a Darcy-Stokes system modeling vuggy porous media, Computational Geosciences 10 (2006) 3, 291-302.
• [2] T. Arbogast, D. Brunson, A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium, Computational Geosciences 1 (2007) 3, 207-218.
• [3] S.G. Beavers, D.D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech. 30 (1967), 197-207.
• [4] J.R. Cannon, G.H. Meyer, Diffusion in a fractured medium, SIAM Journal of Applied Mathematics 20 (1971), 434-448.
• [5] Y. Cao, M. Gunzburger, F. Hua, X. Wang, Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition, Commun. Math. Sci. 8 (2010) 1, 1-25.
• [6] N. Chen, M. Gunzburger, X. Wang, Asymptotic analysis of the differences between the Stokes-Darcy system with different interface conditions and the Stokes-Brinkman system, J. Math. Anal. Appl. 368 (2009) 2, 658-676.
• [7] G.N. Gatica, S. Meddahi, R. Oyarzua, A conforming mixed finite-element method for the coupling of fluid flow with porous media flow, IMA Journal of Numerical Analysis 29 (2009) 1, 86-108.
• [8] P. Grinfeld, G. Strang, The Laplacian eigenvalues of a polygon, Comput. Math. Appl. 48 (2004), 1121-1133.
• [9] P. Grinfeld, G. Strang, Laplace eigenvalues on regular polygons: a series in l/n, J. Math. Anal. Appl. 385 (2012), 1, 135-149.
• [10] M. Higashino, H.G. Stefan, Diffusive boundary layer development above a sediment--water interface, Water Environment Research 76 (2004) 4, 293-300.
• [11] U. Hornung (ed.), Homogenization and Porous Media, vol. 6 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, 1997.
• [12] W.J. Layton, F. Scheiweck, I. Yotov, Coupling fluid flow with porous media flow, SIAM Journal of Numerical Analysis 40 (2003) 6, 2195-2218.
• [13] T. Levy, Fluid flow through an array of fixed particles, International Journal of Engineering Science 21 (1983), 11-23.
• [14] V. Martin, J. Jaffre, J.E. Roberts, Modeling fractures and barriers as interfaces for flow in porous media, SIAM J. Sci. Comput. 26 (2005) 5, 1667-1691.
• [15] F. Morales, R. Showalter, The narrow fracture approximation by channeled flow, J. Math. Anal. Appl. 365 (2010), 320-331.
• [16] F. Morales, R. Showalter, Interface approximation of Darcy flow in a narrow channel, Mathematical Methods in the Applied Sciences 35 (2012), 182-195.
• [17] F.A. Morales, Analysis of a coupled Darcy multiple scale flow model under geometric perturbations of the interface, Journal of Mathematics Research 5 (2013) 4, 11-25.
• [18] F.A. Morales, Homogenization of geological fissured systems with curved non-periodic cracks, Electron. J. Differential Equations 2014 (2014) 189, 1-29.
• [19] A. Paster, G. Dagan, Mixing at the interface between two fluids in porous media: a boundary-layer solution, Journal of Fluid Mechanics 584 (2007), 455-472.
• [20] P.G. Saffman, On the boundary condition at the interface of a porous medium, Studies in Applied Mathematics 1 (1971), 93-101.
• [21] E. Sanchez-Palencia, Nonhomogeneous Media and Vibration Theory, vol. 127 of Lecture Notes in Physics, Springer-Verlag, Berlin, 1980.
• [22] R.E. Showalter, Micro structure Models of Porous Media, [in:] Ulrich Hornung (ed.) Homogenization and Porous Media, vol. 6 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, 1997.
• [23] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Theory, Universitext, Springer, New York, 2007.