  In this work, we introduce an inverse averaging finite element method (IAFEM)
for solving the size-modified Poisson-Nernst-Planck (SMPNP) equations.
Comparing with the classical Poisson-Nernst-Planck (PNP) equations, the SMPNP
equations add a nonlinear term to each of the Nernst-Planck (NP) fluxes to
describe the steric repulsion which can treat multiple nonuniform particle
sizes in simulations. Since the new terms include sums and gradients of ion
concentrations, the nonlinear coupling of SMPNP equations is much stronger than
that of PNP equations. By introducing a generalized Slotboom transform, each of
the size-modified NP equation is transformed into a self-adjoint equation with
exponentially behaved coefficient, which has similar simple form to the
standard NP equation with the Slotboom transformation. This treatment enables
employing our recently developed inverse averaging technique to deal with the
exponential coefficients of the reformulated formulations, featured with
advantages of numerical stability and flux conservation especially in strong
nonlinear and convection-dominated cases. Comparing with previous stabilization
methods, the IAFEM proposed in this paper can still possess the numerical
stability when dealing with convection-dominated problems. And it is more
concise and easier to be numerically implemented. Numerical experiments about a
model problem with analytic solutions are presented to verify the accuracy and
order of IAFEM for SMPNP equations. Studies about the size-effects of a sphere
model and an ion channel system are presented to show that our IAFEM is more
effective and robust than the traditional finite element method (FEM) when
solving SMPNP equations in simulations of biological systems.
