Opuscula Math. 35, no. 4 (2015), 517-546
http://dx.doi.org/10.7494/OpMath.2015.35.4.517

Opuscula Mathematica

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

Fernando A. Morales

Abstract. 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.

Keywords: multiscale coupled systems, interface geometric perturbations, variational formulations, nonlinear dependence.

Mathematics Subject Classification: 58F15, 58F17, 53C35.

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• Fernando A. Morales
• Universidad Nacional de Colombia, Escuela de Matemáticas, Sede Medellín, Calle 59 A No 63-20, Officina 43-106, Medellín, Colombia
• Communicated by P.A. Cojuhari.
• Revised: 2014-10-10.
• Accepted: 2014-10-14.
• Published online: 2015-02-06.