Homogenization of the generalized Poisson–Nernst–Planck problem in a two-phase medium: correctors and estimates

Abstract

The paper provides a rigorous homogenization of the Poisson–Nernst–Planck problem stated in an inhomogeneous domain composed of two, solid and pore, phases. The generalized PNP model is constituted of the Fickian cross-diffusion law coupled with electrostatic and quasi-Fermi electrochemical potentials, and Darcy's flow model. At the interface between two phases inhomogeneous boundary conditions describing electrochemical reactions are considered. The resulting doubly non-linear problem admits discontinuous solutions caused by jumps of field variables. Using an averaged problem and first-order asymptotic correctors, the homogenization procedure gives us an asymptotic expansion of the solution which is justified by residual error estimates.

COMMUNICATED BY:

• Andrey Piatnitski

Keywords:

• Generalized Poisson–Nernst–Planck model
• two-phase interface condition
• homogenization
• periodic unfolding method
• residual error estimate

AMS Subject Classifications:

• 35B27
• 35M10
• 82C24

Acknowledgements

The authors thank two referees for the comments which helped to improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors are supported by the Austrian Science Fund (FWF) Project P26147-N26: ‘Object identification problems: numerical analysis’ (PION). V.A.K. thanks the Austrian Academy of Sciences (OeAW), [the RFBR and JSPS research project 19-51-50004], and A.V.Z. thanks IGDK1754 for partial support.