A Two-Grid Finite Element Method for the Nonlinear Poisson-Boltzmann Equation Based on a Solution Decomposition
Abstract
The nonlinear Poisson--Boltzmann equation (PBE) is a widely used implicit-solvent model for the electrostatics of biomolecules in ionic solvent. Its numerical solution is complicated by the point charges, the dielectric jump across the molecular surface, and the exponential nonlinearity. We study a two-grid finite element method for the PBE built on the solution decomposition u=G+Ψ+Phit of Xie [J.~Comput.~Phys.~275 (2014) 294--309], in which G carries the singularities analytically, Ψ solves a linear interface problem, and Phit solves a semilinear interface problem for an arbitrary number of ionic species. The nonlinear problem for Phit is solved on a coarse mesh only, followed by a single Newton correction on the fine mesh, so that only symmetric positive definite linear systems are solved on the fine mesh. The analysis uses a C2 truncation of the exponential that grows quadratically beyond a level τ; for τ above an explicit bound the truncated problem has the same solution as the original one, and no discrete maximum principle or mesh-angle condition is needed. Under standard piecewise regularity assumptions we prove quasi-optimal energy and L2 estimates and an L4 estimate for the finite element approximation of Phit, and for the two-grid solution we prove enormu−utg≤ C(hs−1+H2(s−1)+t/2) in three dimensions, where t=min{s,σ}−1. Whenever the dual problem is at least as regular as the primal one (σ≥ s), the coarse mesh can be chosen as H∼ h2/5 without loss of accuracy. A manufactured-solution test confirms the predicted rates and the quadratic dependence of the two-grid correction on the coarse error.
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BibTeX
@misc{chao2026twogrid,
title = {{A Two-Grid Finite Element Method for the Nonlinear Poisson-Boltzmann Equation Based on a Solution Decomposition}},
author = {Zhen Chao and Jingchun Chen and Cong He},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34661},
eprint = {2610.34661},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34661},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Two-Grid Finite Element Method for the Nonlinear Poisson-Boltzmann Equation Based on a Solution Decomposition
AU - Zhen Chao
AU - Jingchun Chen
AU - Cong He
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34661
UR - https://arxiv.org/abs/2610.34661
AB - The nonlinear Poisson--Boltzmann equation (PBE) is a widely used implicit-solvent model for the electrostatics of biomolecules in ionic solvent. Its numerical solution is complicated by the point charges, the dielectric jump across the molecular surface, and the exponential nonlinearity. We study a two-grid finite element method for the PBE built on the solution decomposition u=G+Ψ+Phit of Xie [J.~Comput.~Phys.~275 (2014) 294--309], in which G carries the singularities analytically, Ψ solves a linear interface problem, and Phit solves a semilinear interface problem for an arbitrary number of ionic species. The nonlinear problem for Phit is solved on a coarse mesh only, followed by a single Newton correction on the fine mesh, so that only symmetric positive definite linear systems are solved on the fine mesh. The analysis uses a C^2 truncation of the exponential that grows quadratically beyond a level τ; for τ above an explicit bound the truncated problem has the same solution as the original one, and no discrete maximum principle or mesh-angle condition is needed. Under standard piecewise regularity assumptions we prove quasi-optimal energy and L^2 estimates and an L^4 estimate for the finite element approximation of Phit, and for the two-grid solution we prove enormu−utg≤ C(h^s−1+H^2(s−1)+t/2) in three dimensions, where t=min{s,σ}−1. Whenever the dual problem is at least as regular as the primal one (σ≥ s), the coarse mesh can be chosen as H∼ h^2/5 without loss of accuracy. A manufactured-solution test confirms the predicted rates and the quadratic dependence of the two-grid correction on the coarse error.
ER -