Finite-Difference Approximations and Local Algorithm for the Poisson and Poisson-Boltzmann Electrostatics

Abstract

We study finite-difference approximations of both Poisson and Poisson-Boltzmann (PB) electrostatic energy functionals for periodic structures constrained by Gauss' law and a class of local algorithms for minimizing the finite-difference discretization of such functionals. The variable of Poisson energy is the vector field of electric displacement and that for the PB energy consists of an electric displacement and ionic concentrations. The displacement is discretized at midpoints of edges of grid boxes while the concentrations are discretize at grid points. The local algorithm is an iteration over all the grid boxes that locally minimizes the energy on each grid box, keeping Gauss' law satisfied. We prove that the energy functionals admit unique minimizers that are solutions to the corresponding Poisson's and charge-conserved PB equation, respectively. Local equilibrium conditions are identified to characterize the finite-difference minimizers of the discretized energy functionals. These conditions are the curl free for the Poisson case and the discrete Boltzmann distributions for the PB case, respectively. Next, we obtain the uniform bound with respect to the grid size h and O(h2)-error estimates in maximum norm for the finite-difference minimizers. The local algorithms are detailed, and a new local algorithm with shift is proposed to treat the general case of a variable coefficient for the Poisson energy. We prove the convergence of all these local algorithms, using the characterization of the finite-difference minimizers. Finally, we present numerical tests to demonstrate the results of our analysis.

Figures (7)

• Figure 5.1: (Left) The grid box $B_{i, j, k} = (i, j, k)+[0,1]^3.$ (Right) The grid face of box $B_{i, j, k}$ with vertices $P = (i, j, k)$, $Q = (i+1, j, k)$, $R = (i+1, j+1, k)$, and $S = (i, j+1, k).$ The perturbations $\alpha$, $\beta$, $\gamma$ and $\delta$ of $u$ and $v$ with subscript, the corresponding components of the displacement $D$, are to be determined.
• Figure 6.1: The discrete energy (a), $L^2$-error (b), and $L^{\infty}$-error (c) for the displacement $D^{(k)}_h$ vs. the iteration step $k$ in the local algorithm for Test 1.
• Figure 6.2: Log-log plots of the $L^2$-error (a) and the $L^{\infty}$-error (b) for the approximation $D_h$ of the displacement $D$ (indicated by $D$) and the reconstructed approximation $E_h:=m_h[D_h]/\varepsilon$ of the electric field $E := -\nabla \phi$ (indicated by $E$) for Test 1. The blue dashed lines are reference lines indicating the $O(h^2)$ convergence rate.
• Figure 6.3: The discrete energy (a), $L^2$-error (b), and $L^{\infty}$-error (c) for the displacement $D^{(k)}_h$ vs. the iteration step $k$ in the local algorithm with shift for Test 2.
• Figure 6.4: Log-log plots of the $L^2$-error (a) and the $L^{\infty}$-error (b) for the approximation $D_h$ of the displacement $D$ (marked $D$) and the reconstructed approximation $E_h:=m_h[D_h]/\varepsilon$ of the electric field $E := -\nabla \phi$ (marked $E$) for Test 2. The blue dashed lines (marked Ref) are reference lines indicating the $O(h^2)$ convergence rate.

Theorems & Definitions (52)

• Theorem 2.1
• proof
• Lemma 2.1
• proof
• Definition 2.1
• Theorem 2.2
• Remark 2.1
• proof : Proof of Theorem \ref{['t:CCPBenergy']}
• Lemma 2.2
• proof