Nitsche method for the Stokes-Poisson-Boltzmann equation with Navier slip boundary condition
This work provides a rigorous numerical analysis for a coupled fluid-electrostatic system with slip boundaries, which is important for microfluidic and biological applications, but the contribution is incremental as it extends existing Nitsche techniques to a specific coupled PDE system.
The paper develops a Nitsche-based finite element method for the Stokes-Poisson-Boltzmann equations with Navier slip boundary conditions, proving well-posedness, optimal convergence rates, and reliable a posteriori error estimators, with numerical experiments confirming the theory.
We study the Stokes--Poisson--Boltzmann equations with Dirichlet and Navier boundary conditions. The system consists of the incompressible Stokes equations coupled with a nonlinear Poisson--Boltzmann equation through electrostatic forcing and convective transport effects. To handle the Navier boundary conditions in a unified framework, we employ Nitsche's method for their weak imposition within a conforming finite element setting. We derive a consistent and stable discrete formulation and establish the well-posedness of the resulting problem. By carefully choosing the penalty parameters, the bilinear form is shown to be coercive and continuous. A priori error estimates are proved in the natural energy norms, yielding optimal-order convergence under suitable regularity assumptions. Furthermore, we develop residual-based a posteriori error estimators that incorporate element residuals, inter-element jump residuals, and boundary residuals arising from the Nitsche formulation. The estimators are shown to be reliable and locally efficient. Numerical experiments confirm the theoretical results and demonstrate the robustness and accuracy of the proposed method for the Stokes--Poisson--Boltzmann system.