Energy-norm optimal convergence of a structure-preserving predictor–projection Fourier–Galerkin method for the Brinkman–Poisson–Nernst–Planck system

In this paper, a fully decoupled, structure-preserving Fourier–Galerkin method is developed and analyzed for the time-dependent Brinkman–Poisson–Nernst–Planck (BPNP) system, which models monovalent ion electrodiffusion in an incompressible porous medium. The scheme combines an entropy-gradient flow formulation for the Poisson–Nernst–Planck block with a predictor–projection method for the Brinkman momentum equation, yielding three sequentially solved subproblems: a convex minimization for ionic concentrations, a Poisson equation for the electric potential, and a linear Brinkman predictor followed by an incompressibility correction. The method guarantees exact mass conservation for each ion species, pointwise positivity of concentrations, and unconditional discrete free energy dissipation for any time step. For smooth and strictly positive initial data, we prove first-order accuracy in time and optimal convergence rates in space in the energy norm, for the Fourier–Galerkin discretization on the periodic box. A key analytical technique is the construction of high-order consistent correctors that ensure uniform L∞ bounds for the discrete concentrations, enabling a bootstrap argument to close the nonlinear error estimates. Numerical experiments based on Fourier modes validate the theoretical results, including positivity preservation, mass conservation, monotonic energy decay, and the predicted convergence rates. As a starting point for more complex systems, the proposed decoupled framework and its analytical tools can be extended to BPNP–Navier–Stokes coupling problems arising in ion channels and cellular environments, which will be pursued in future work.