Adaptive and Iterative Methods for Simulations of Nanopores with the PNP-Stokes Equations

TL;DR

This paper introduces an advanced 3D finite element solver for coupled PNP-Stokes equations, employing goal-oriented adaptive mesh refinement and novel linearization schemes, to simulate nanopore sensors with improved efficiency and physical accuracy.

Contribution

It develops a new adaptive mesh refinement approach using the Poisson-Boltzmann equation and proposes three linearization schemes for the nonlinear system, enhancing simulation efficiency and accuracy.

Findings

Adaptive mesh refinement improves computational efficiency.

Segregated linearization schemes outperform naive Newton's method.

Point-size molecule model misses key physical effects.

Abstract

We present a 3D finite element solver for the nonlinear Poisson-Nernst-Planck (PNP) equations for electrodiffusion, coupled to the Stokes system of fluid dynamics. The model serves as a building block for the simulation of macromolecule dynamics inside nanopore sensors. We add to existing numerical approaches by deploying goal-oriented adaptive mesh refinement. To reduce the computation overhead of mesh adaptivity, our error estimator uses the much cheaper Poisson-Boltzmann equation as a simplified model, which is justified on heuristic grounds but shown to work well in practice. To address the nonlinearity in the full PNP-Stokes system, three different linearization schemes are proposed and investigated, with two segregated iterative approaches both outperforming a naive application of Newton's method. Numerical experiments are reported on a real-world nanopore sensor geometry. We also investigate two different models for the interaction of target molecules with the nanopore sensor through the PNP-Stokes equations. In one model, the molecule is of finite size and is explicitly built into the geometry; while in the other, the molecule is located at a single point and only modeled implicitly -- after solution of the system -- which is computationally favorable. We compare the resulting force profiles of the electric and velocity fields acting on the molecule, and conclude that the point-size model fails to capture important physical effects such as the dependence of charge selectivity of the sensor on the molecule radius.