An extended finite element method for the Nernst-Planck-Poisson equations

The Nernst-Planck-Poisson (NPP) system of equations is used to study the ion diffusion mechanism in materials, commonly used in storage batteries and solid oxide fuel cells. In particular, NPP equations are used to predict the concentration of charged defects and electric potential. Depending on the size of the computational domain, the distributions exhibit steep gradients near the boundaries. The traditional finite element method, when employed, requires extremely refined mesh to capture the steep gradient as they employ simple polynomials. To alleviate the mesh dependence, in this work, we propose to augment the traditional finite element approximation space with a suitable ansatz to capture the steep gradient within the framework of the extended finite element method. The robustness and the accuracy of the proposed framework is demonstrated by comparing it with an overkill finite element solution.