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Seminar Numerische Mathematik
WIAS Berlin
M. Schmuck (Imperial College London)
Upscaling of ionic transport equations in strongly heterogeneous media and finite element approximations
Programm / Abstract:
We consider the well-accepted Nernst-Planck-Poisson equations [6] for the description of ionic transport and electrokinetic phenomena such as electro-phoresis and -osmosis. Applications range from designing microfluidic devices, energy storage devices, and semiconductors to emulating communication in biological cells by synthetic nanopores. Based on this classical description, we derive a new effective macroscopic set of equations [1,2,3] which describe binary symmetric electrolytes in porous and strongly heterogeneous media. Heterogeneous materials naturally induce corrected transport parameters which we call ''material tensors''. Our systematic and well-accepted upscaling strategy by homogenization gives a reliable understanding on the influence of the micro-geometry on the macroscale. The new equations provide an essential computational advantage by a strong reduction of the degrees of freedom in comparison to the classical description which requires a full resolution of the pore geometry and hence a high-dimensional resolution of the problem. For the classical descritpion, we briefly present a linear finite element scheme [4,5] that is reliable in the sense that discrete solutions conserve mass, are non-negative and bounded, and converge towards weak solutions. Finally, we can qualitatively and quantitatively characterize the suitability of the new upscaled equations for specific applications by error estimates [2] which compare the exact microscopic solution with the solution of the new effective transport equations.
References:
[1] M. Schmuck, A new upscaled Poisson-Nernst-Planck system for strongly oscillating potentials, J. Math. Phys. 54:021504 (2013).
[2] M. Schmuck, First error bounds for the porous media approximation of the Poisson-Nernst-Planck equations, Z. angew. Math. Mech. 92:304-319 (2012).
[3] M. Schmuck and P. Berg, Homogenization of a catalyst layer model for periodically distributed pore geometries in PEM fuel cells, Appl. Math. Res. Express. 2013(1):57-78 (2012).
[4] A. Prohl and M. Schmuck, Convergent finite element discretizations of the Navier-Stokes-Nernst-Planck-Poisson system, ESAIM, Math. Model. Numer. Anal. 44(3):531-571 (2010).
[5] A. Prohl, and M. Schmuck, Convergent discretizations for the Nernst-Planck-Poisson system, Num. Math. 111 (4):591-630 (2009).
[6] M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Mod. Meth. Appl. S. 19(6):993- 1015 (2009).
Zeit:
am Donnerstag den 18. April 2013 um 14:00
Ort:
Weierstraß-Institut
Mohrenstr. 39
10117 Berlin
Erhard-Schmidt-Hörsaal Erdgeschoss
eingetragen von lawrenz(Marion.Lawrenz@wias-berlin.de, 030 20372566)