Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

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

TL;DR

The paper develops an equal-order virtual element method for the coupled Stokes--Poisson--Boltzmann equations on general polygonal meshes, introducing a residual-based pressure stabilization that reformulates the Laplacian drag as a transport-velocity-potential term to avoid second-order derivatives. Existence and uniqueness are proven via fixed-point arguments under a small-data regime, and optimal a priori error estimates of order $\mathcal{O}(h^k)$ are established in the energy norm for polynomial degrees $k\ge1$. Numerical experiments on convex, non-convex, and hanging-node meshes confirm the predicted convergence across velocity, pressure, and potential, and demonstrate the method’s applicability to electro-osmotic flows in nanopore geometries. Compared to Taylor--Hood FEM, the VE framework offers equal-order approximations with uniform DOFs, natural handling of polygonal elements and hanging nodes, and notable reductions in degrees of freedom, while maintaining robust stability and accuracy for complex electrokinetic simulations.

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.

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

TL;DR

The paper develops an equal-order virtual element method for the coupled Stokes--Poisson--Boltzmann equations on general polygonal meshes, introducing a residual-based pressure stabilization that reformulates the Laplacian drag as a transport-velocity-potential term to avoid second-order derivatives. Existence and uniqueness are proven via fixed-point arguments under a small-data regime, and optimal a priori error estimates of order are established in the energy norm for polynomial degrees . Numerical experiments on convex, non-convex, and hanging-node meshes confirm the predicted convergence across velocity, pressure, and potential, and demonstrate the method’s applicability to electro-osmotic flows in nanopore geometries. Compared to Taylor--Hood FEM, the VE framework offers equal-order approximations with uniform DOFs, natural handling of polygonal elements and hanging nodes, and notable reductions in degrees of freedom, while maintaining robust stability and accuracy for complex electrokinetic simulations.

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 for approximation degree . 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.
Paper Structure (21 sections, 18 theorems, 155 equations, 6 figures, 5 tables)

This paper contains 21 sections, 18 theorems, 155 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Outline
  3. Model problem
  4. Weak formulation and well-posedness
  5. Well-posedness of the decoupled problems
  6. Well-posedness of the coupled problem
  7. Stabilized virtual element framework
  8. Pressure stabilizations
  9. Projectors and virtual element spaces
  10. Discrete forms and stabilized virtual element problem
  11. Theoretical analysis
  12. Stability of discrete decoupled problems
  13. Well-posedness of the stabilized virtual element problem
  14. A priori error estimate
  15. Discussion of the smallness condition
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $\psi\in\mathlarger \Phi$ be such that $|\space|\nabla\psi|\space|_{\Omega}\leq\frac{\mu}{2C_{1 \hookrightarrow4}^2E^\ast}$. Then, the decoupled problem variation1 has a unique solution $(\mathbf{u},p)\in\mathbf{V}\times Q$. Furthermore, it also satisfies

Figures (6)

  • Figure 1: Convex and non-convex domains with general polygons including hanging nodes $\Omega_4$.
  • Figure 2: [Example 3] Computational domains with their geometry.
  • Figure 3: [Example 3] Snapshots of flow dynamics of the electrically charged fluid in a nanosensor: discrete velocity magnitude, pressure, and potential for VEM order $k=1$ with $\mathbf{E}=[0.1, -0.1]^t$
  • Figure 4: [Example 3] Snapshots of the flow dynamics of the electrically charged fluid in a nanosensor: discrete velocity magnitude, pressure, and potential for VEM order $k=2$ with $\mathbf{E}=[0.1, -0.1]^t$
  • Figure 5: [Example 3] Snapshots of the flow dynamics of the electrically charged fluid in a nanosensor: discrete velocity magnitude, pressure, and potential for VEM order $k=2$ with $\mathbf{E}=[1, -1]^t$.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Proposition 2.5
  • Remark 3.1
  • ...and 24 more