First error bounds for the porous media approximation of the Poisson‐Nernst‐Planck equations
Markus Schmuck · ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik · 2012
Abstract 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 L2‐norm and the electric potential measured in H1‐norm, are of order O(s1/2).