Stabilized Finite Element Method for Poisson Nernst-Planck Equations with Steric Effects for Ion Transport

Abidha Monica Gwecho; Wang Shu; Onyango Thomas Mboya

doi:https://doi.org/10.14445/22315373/IJMTT-V68I4P515

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 68 | Issue 4 | Year 2022 | Article Id. IJMTT-V68I4P515 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I4P515

Stabilized Finite Element Method for Poisson Nernst-Planck Equations with Steric Effects for Ion Transport

Abidha Monica Gwecho, Wang Shu, Onyango Thomas Mboya

Received	Revised	Accepted
19 Mar 2022	21 Apr 2022	26 Apr 2022

Citation :

Abidha Monica Gwecho, Wang Shu, Onyango Thomas Mboya, "Stabilized Finite Element Method for Poisson Nernst-Planck Equations with Steric Effects for Ion Transport," International Journal of Mathematics Trends and Technology (IJMTT), vol. 68, no. 4, pp. 94-101, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I4P515

Abstract

In this paper, Nernst-Planck(NP) equation for ion fluxes that uses Lennard Jones(LJ) potential to incorporate finite-size effects in terms of hard sphere repulsion was coupled with Poisson equations to form modified PNP(mPNP) system of equations. These coupled equations were then discretized using Galerkin finite element Method(GFEM) approach based on Taylor-hood elements with regular rectangular sub-domains. However, this method resulted into numerical oscillations in the approximate solution, necessitating stabilization to obtain desired results. Consequently, Stream-Upwind Petrov-Galerkin(SUPG)method which adds a mesh dependent term to the FEM together with iterative linearization was adopted resulting into a stable numerical scheme. The resulting linear system of equations was solved iteratively using preconditioned conjugate gradient(PCG) scheme to speed up convergence, where potential consistently updated the concentration components. Concentration profiles of two ion species under varied steric effects for mPNP and PNP were compared and analyzed.

Keywords

Modified PNP, Lennard Jones, Hard-sphere, SUPG, Finite-size effects.

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