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arXiv · 2602.16538

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

Sudheer Mishra·Sundararajan Natarajan·E. Natarajan·Gianmarco Manzini
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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.

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Sudheer Mishra, Sundararajan Natarajan, E. Natarajan, Gianmarco Manzini. 2026-07-28. A higher order pressure-stabilized virtual element formulation for the Stokes-Poisson-Boltzmann equations. https://arxiv.org/abs/2602.16538

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