A Two-Grid Finite Element Method for the Nonlinear Poisson-Boltzmann Equation Based on a Solution Decomposition

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 $\enorm{u-\utg}\le C\big(h^{s-1}+H^{2(s-1)+t/2}\big)$ in three dimensions, where $t=\min\{s,σ\}-1$. Whenever the dual problem is at least as regular as the primal one ($σ\ge s$), the coarse mesh can be chosen as $H\sim h^{2/5}$ 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.