Finite element method for Poisson equations with the Wang-Ball element

In electrostatic field research, the Poisson equation is the core equation describing the relationship between electric potential and charge distribution. The finite element method (FEM) is an analytical engineering tool for accurately calculating various physical quantities in the electrostatic field. In this paper, we employ the Wang-Ball basis functions to construct the trial and test function spaces of FEM for solving the Poisson equation. In addition, we provide an error analysis based on the Wang-Ball operator. Several examples with different electrostatics backgrounds are also given to substantiate the effectiveness of this method. Furthermore, numerical results show that Wang-Ball elements work well for degree $k>3$.