Multidomain Chebyshev pseudo-spectral method applied to the Poisson–Boltzmann equation for two parallel plates

Abstract

We address a boundary-value problem involving a Poisson–Boltzmann equation that models the electrostatic potential of a channel formed by parallel plates with an electrolyte solution confined between the plates. We show the existence and uniqueness of solution to the problem, with special (particular) solutions as bounds, namely, a Debye–Hückel type solution as lower bound and a Gouy–Chapman type solution as upper bound. Our results are based on the maximum principle for elliptic equations and are useful for characterizing the behavior of the solutions. Also, we introduce a numerical scheme based on the Chebyshev pseudo-spectral method to calculate approximate solutions. This method is applied in conjunction with a multidomain procedure that attempts to capture the dramatic exponential increase/decay of the solution near the plates.

Appendix

Here we present without proof the maximum principle for boundary-value problems for second-order nonlinear elliptic equations as stated in [28, Theorem 22, p. 48].

Theorem 4

Let u(x) the solution of the boundary-value problem

$$\begin{aligned}&u''+H(x,u,u')=0,\quad a<x<b \end{aligned}$$

(88)

$$\begin{aligned}&u(a)=\zeta _1,\quad u(b)=\zeta _2. \end{aligned}$$

(89)

Suppose that H, $\dfrac{\partial H}{\partial y}$, $\dfrac{\partial H}{\partial z}$ are continuous and that $\dfrac{\partial H}{\partial y}\le 0$. If $z_1(x)$ satisfies

$$\begin{aligned}&z_1''+H(x,z_1,z_1')\le 0,\quad a<x<b \end{aligned}$$

(90)

$$\begin{aligned}&z_1(a)\ge \zeta _1,\quad z_1(b)\ge \gamma _2 \end{aligned}$$

(91)

and if $z_2(x)$ satisfies

$$\begin{aligned}&z_2''+H(x,z_2,z_2')\ge 0,\quad a<x<b \end{aligned}$$

(92)

$$\begin{aligned}&z_2(a)\le \gamma _1,\quad z_2(b)\le \gamma _2, \end{aligned}$$

(93)

thus the upper and lower bounds

$$\begin{aligned} z_2(x)\le u(x)\le z_1(x) \end{aligned}$$

(94)

are valid.