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.
Nanopore sensors have emerged as transformative tools for single-molecule analysis, DNA sequencing, and ion channel studies [1]. The fundamental physics governing these devices involves coupled fluid-electrostatic interactions: ionic transport through nanoscale pores is driven by both pressure gradients and applied electric fields, while electrostatic potentials arise from charge distributions in thin electrical double layers near solid boundaries. Accurate numerical simulation of these electrokinetic phenomena requires solving the Stokes equations coupled with the Poisson-Boltzmann equation, and the geometric complexity of realistic nanopore structures-including irregular pore shapes, multiple obstacles, and interfaces requiring adaptive mesh refinement-poses significant challenges for conventional numerical methods.
Standard finite element methods (FEM) for the Stokes-Poisson-Boltzmann (SPB) system typically employ mixed approximation spaces to satisfy the discrete inf-sup condition. For instance, Taylor-Hood P k+1 /P k elements [2] use higher-order velocity approximations to ensure stability. While theoretically sound, these approaches face practical limitations: (i) implementation complexity of mixed finite element spaces with different approximation orders, (ii) difficulty handling hanging nodes arising from adaptive mesh refinement, and (iii) restricted mesh flexibility when approximating domains with irregular boundaries. Virtual element methods (VEM) offer an attractive alternative by enabling equal-order approximations on general polygonal meshes, with a projection-based framework that naturally handles arbitrary element shapes and hanging nodes.
Recent work by AlSohaim, Ruiz-Baier, and Villa-Fuentes [2] developed a finite element formulation for the SPB system using Taylor-Hood spaces on triangular meshes, establishing theoretical foundations through fixed-point analysis and deriving optimal error estimates. While this work provides important theoretical results, the geometric constraints of conforming triangular meshes and the complexity of mixed-order implementations motivate exploration of alternative approaches. In particular, equal-order approximations and the polytopal flexibility offered by VEM remain unexplored for this problem class.
Electrokinetic phenomena in charged fluids play a significant role in micro-scale and nano-scale transport processes [3,4,5], with applications spanning nanopore design for DNA sequencing, ion transport modeling in water purification systems [6], and microfluidic device development [7]. Among these, electro-osmotic flow-arising from the interaction between an externally applied electric field and charged species within thin electric double layers-represents one of the most efficient mechanisms for inducing fluid motion without mechanical actuation. In this work, fluid motion is modeled using the Stokes equations with a forcing term dependent on the electrolyte charge and the applied electric field. The fluid velocities considered are not sufficiently strong to affect the electrostatic potential distribution within the double layer; consequently, the electric charge density can be directly related to the potential through the Poisson-Boltzmann equation, supplemented by an advection term coupling the fluid velocity and potential.
Concerning the Poisson-Boltzmann equation (PBE), finite element formulations have been discussed in [8,9,10], with extension to general polygons in [11]. Following [2], we consider the regularized form of this equation, which preserves the nonlinearity characterized by a hyperbolic sine function but omits the distributional Dirac forcing term. We also incorporate the convection term coupling fluid velocity and potential, and impose a restriction on the functional space requiring the double layer potential field to be bounded as discussed in [10].
Polytopal methods have attracted considerable attention because they provide flexibility in dealing with complex geometries and irregular interfaces, where standard methods like FEM or the finite difference method (FDM) might be computationally costly. To address the challenge that extending FEM to polytopal meshes requires explicit construction of shape functions, Beirão da Veiga et al. [12] introduced the virtual element method (VEM), which handles arbitrary polygonal/polyhedral meshes without explicit basis function formulation. Instead, VEM requires only suitable choices of degrees of freedom to compute the discrete formulation, enabling easy extension to higher-order approximations while maintaining robustness under general mesh types. VEM has been applied to elasticity [13,14], Oseen problem [15,16], Stokes and Navier-Stokes problems [17,18,19,20,21,22], Maxwell equation [23], plate bending [24], crack propagation [25,26], optimization [27,28], and magnetostatics [29]. For the Stokes problem, Guo and Feng [30] introduced a projection-based stabilized VEM using equal-order velocity-p
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