Use and comparative assessment of the CVFEM method for Poisson-Boltzmann and Poisson-Nernst-Planck three dimensional simulations of impedimetric nano-biosensors operated in the DC and AC small signal regimes

We specialize the Control Volume Finite Element Method (CVFEM) for the solution of the Poisson-Boltzmann and Poisson-Nernst-Planck (also known as Poisson-drift-diffusion) system of equations on unstructured 3D meshes describing nanoelectronic biosensors operated in the DC and AC small signal regimes. We provide the exact analytical expressions for volume and surface integrals derived by means of a linear coordinate transformation and show that they are both accurate and efficient, especially on coarse grids. Being of great importance for the chosen application, the conservation property is investigated and we show that, for the CVFEM to be conservative, the calculations on the boundary have to be performed with special care. CVFEM is carefully compared to the Galerkin Finite Element Method (GFEM) from the point of view of the underlying theory, implementation and solution calculation. The simulation tool is used to evaluate the response of a real nanoelectrode-based biosensor array to the introduction of small nanoparticles.