Demonstrates a local algorithm minimizing electrostatic energy in finite-difference models, suggesting effective techniques for computational physics.
We study finite-difference approximations of the Poisson–Boltzmann (PB) electrostatic energy functional of ionic concentrations and electric displacements constrained by Gauss’ law and the ionic mass conservation, and a class of local algorithms for minimizing the finite-difference discretized such energy functional. We prove that the discrete Boltzmann distributions characterize the finite-difference minimizer and obtain the uniform bounds and optimal error estimates in maximum norm for such a minimizer. The local algorithm is an iteration over all the grid boxes that locally minimizes the energy by updating the concentrations and displacement one grid box at a time, keeping Gauss’ law and the mass conservation satisfied. A new local algorithm with a shift is constructed for minimizing the Poisson electrostatic energy (the part of the PB energy without ionic concentrations) with a variable dielectric coefficient. We prove the convergence of these local algorithms and present numerical tests to demonstrate the results of our analysis.
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Li et al. (2026) studied this question.
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