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First error bounds for the porous media approximation of the Poisson‐Nernst‐Planck equations
Author(s) -
Schmuck M.
Publication year - 2012
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201100003
Subject(s) - nernst equation , planck , porous medium , electrolyte , norm (philosophy) , poisson distribution , approximation error , convergence (economics) , uniform norm , physics , uniqueness , homogeneous , mathematics , mathematical analysis , statistical physics , porosity , materials science , quantum mechanics , electrode , statistics , political science , law , economics , composite material , economic growth
We study the well‐accepted Poisson‐Nernst‐Planck equations modeling transport of charged particles. By formal multiscale expansions we rederive the porous media formulation obtained by two‐scale convergence in [42, 43]. The main result is the derivation of the error which occurs after replacing a highly heterogeneous solid‐electrolyte composite by a homogeneous one. The derived estimates show that the approximation errors for both, the ion densities quantified in L 2 ‐norm and the electric potential measured in H 1 ‐norm, are of order O(s 1/2 ).
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