Nitsche method for the Stokes-Poisson-Boltzmann equation with Navier slip boundary condition

TL;DR

This paper develops a finite element Nitsche method for the coupled Stokes-Poisson-Boltzmann system with Navier boundary conditions, providing a stable, accurate, and efficient numerical approach with proven error estimates.

Contribution

It introduces a unified Nitsche-based finite element framework for Navier boundary conditions in the Stokes-Poisson-Boltzmann system, including stability analysis and error estimation.

Findings

Optimal-order convergence under regularity assumptions

Residual-based a posteriori error estimators are reliable and efficient

Numerical experiments confirm robustness and accuracy

Abstract

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.