YunKyong Hyon

New Poisson–Boltzmann type equations: one-dimensional solutions

The Poisson–Boltzmann (PB) equation is conventionally used to model the equilibrium of bulk ionic species in different media and solvents. In this paper we study a new Poisson–Boltzmann type (PB_n) equation with a small dielectric parameter 2 and non-local nonlinearity which takes into consideration the preservation of the total amount of each individual ion. This equation can be derived from the original Poisson–Nernst–Planck system. Under Robin-type boundary conditions with various coefficient scales, we demonstrate the asymptotic behaviours of one-dimensional solutions of PB_n equations as the parameter approaches zero. In particular, we show that in case of electroneutrality, i.e. α = β, solutions of 1D PB_n equations have a similar asymptotic behaviour as those of 1D PB equations. However, as α ≠ β (non-electroneutrality), solutions of 1D PB_n equations may have blow-up behaviour which cannot be found in 1D PB equations. Such a difference between 1D PB and PB_n equations can also be verified by numerical simulations.

Superconvergence of new mixed finite element spaces

In this paper we prove some superconvergence of a new family of mixed finite element spaces of higher order which we introduced in [ETNA, Vol. 37, pp. 189-201, 2010]. Among all the mixed finite element spaces having an optimal order of convergence on quadrilateral grids, this space has the smallest unknowns. However, the scalar variable is only suboptimal in general; thus we have employed a post-processing technique for the scalar variable. As a byproduct, we have obtained a superconvergence on a rectangular grid. The superconvergence of a velocity variable naturally holds and can be shown by a minor modification of existing theory, but that of a scalar variable requires a new technique, especially for k=1. Numerical experiments are provided to support the theory.