We propose and compare two classes of convergent finite element based approximations of the nonstationary Nernst–Planck–Poisson equations, whose constructions are motivated from energy versus entropy decay properties for the limiting system. Solutions of both schemes converge to weak solutions of the limiting problem for discretization parameters tending to zero. Our main focus is to study qualitative properties for the different approaches at finite discretization scales, like conservation of mass, non-negativity, discrete maximum principle, decay of discrete energies, and entropies to study long-time asymptotics.
Mathematics Subject Classification (2000)
65N30 35L60 35L65
This is a preview of subscription content, log in to check access.
Gajewski H, Gärtner K.: On the discretization of van Roosbroeck’s equations with magnetic field. Z. Angew. Math. Mech. 76, 247–264 (1996)zbMATHCrossRefGoogle Scholar
Grün G., Rumpf M.: Nonnegativity preserving convergent schemes for the thin film equation. J. Math. Anal. Appl. 87, 113–152 (2000)zbMATHGoogle Scholar
Heywood J.G., Rannacher R.: Finite element approximation of the non-stationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
Jerome J.W.: Analysis of Charge Transport. Springer, Berlin (1996)Google Scholar
Jüngel A., Peng Y.J.: A discretization scheme for a quasi-hydrodynamic semiconductor model. Math. Mod. Methods Appl. Sci. 7, 935–955 (1997)zbMATHCrossRefGoogle Scholar
Korotiv S., Krizek M., Neittaanmäki P.: Weakened acute type condition for tetrahedral triangulations and the discrete maximum prinicple. Math. Comp. 70, 107–119 (2001)CrossRefMathSciNetGoogle Scholar
Markovitch P.A.: The Stationary Semiconductor Device Equations. Springer, Wien (1986)Google Scholar