A Finite Element Block Modified Backward Euler Method For Solving A One-Dimensional Poisson-Nernst-Planck Ion Channel Model

Mr. Silas Winnemoeller
University of Wisconsin-Milwaukee

In this thesis, a finite element block modified backward Euler method is introduced to solve a one-dimensional Poisson-Nernst-Planck ion channel (1D PNPic) model. This model is defined as a system of time-dependent nonlinear partial differential equations, called Poisson-Nernst equations and Poisson equation, describing the transport of charged ionic species across a cell membrane via an ion channel pore. For an electrolyte with n ionic species, its numerical solution gives a prediction to n ionic concentration functions and an electrostatic potential function. However, solving the 1DPNPic model numerically is challenging due to the model’s strong nonlinearity and numerical stability issues. To address the numerical stability issues, the traditional backward Euler implicit time scheme is often selected to solve the 1DPNPic model but it may be too costly to be practical in application since it has to solve a system of n + 1 strongly nonlinear partial differential equations at each time step. Hence, its modification becomes necessary to reduce its computing cost while retaining its numerical stability properly. In this thesis, the new method is constructed by semi-discretization and finite element techniques such that its each time iteration only involves calculation within two blocks with each block only containing two linear differential equations. Consequently, the new method can reduce the
computing cost of the Euler scheme sharply. In this thesis, the new method is implemented as a software package in Python based on the finite element library from the FEniCS project. Numerical tests are then done for an electrolyte with two ionic species, demonstrating the convergence and high performance of the new method.

Advisor: Prof. Dexuan Xie

Committee Members:
Profs. Lei Wang, Vincent Larson, and Dexuan Xie