A higher order pressure-stabilized virtual element formulation for the Stokes-Poisson-Boltzmann equations

TL;DR

This paper introduces a novel virtual element method with pressure stabilization for coupled fluid-electrostatic equations, enabling accurate simulations on complex polygonal meshes with theoretical guarantees and practical validation.

Contribution

It develops a residual-based pressure stabilization scheme for the Stokes-Poisson-Boltzmann system using virtual elements, allowing flexible meshing without remeshing or special treatments.

Findings

Achieves optimal convergence rates of order h^k in energy norm.

Successfully handles complex polygonal meshes, including distorted and non-convex elements.

Demonstrates practical applicability in electro-osmotic flow simulations with complex geometries.

Abstract

Electrokinetic phenomena in nanopore sensors and microfluidic devices require accurate simulation of coupled fluid-electrostatic interactions in geometrically complex domains with irregular boundaries and adaptive mesh refinement. We develop an equal-order virtual element method for the Stokes--Poisson--Boltzmann equations that naturally handles general polygonal meshes, including meshes with hanging nodes, without requiring special treatment or remeshing. The key innovation is a residual-based pressure stabilization scheme derived by reformulating the Laplacian drag force in the momentum equation as a weighted advection term involving the nonlinear Poisson--Boltzmann equation, thereby eliminating second-order derivative terms while maintaining theoretical rigor. Well-posedness of the coupled stabilized problem is established using the Banach and Brouwer fixed-point theorems under sufficiently small data assumptions, and optimal a priori error estimates are derived in the energy norm with convergence rates of order $\mathcal{O}(h^k)$ for approximation degree $k \geq 1$. Numerical experiments on diverse polygonal meshes -- including distorted elements, non-convex polygons, Voronoi tessellations, and configurations with hanging nodes -- confirm optimal convergence rates, validating theoretical predictions. Applications to electro-osmotic flows in nanopore sensors with complex obstacle geometries illustrate the method's practical utility for engineering simulations. Compared to Taylor--Hood finite element formulations, the equal-order approach simplifies implementation through uniform polynomial treatment of all fields and offers native support for general polygonal elements.