Solving singularities in electrostatics with high-order FEM

We propose a method to discretize the Poisson equation accurately with nonsmooth areas. This method can simulate the discontinuity of the electric field correctly and has good efficiency compared with the adaptive mesh refinement method. Two types of high-order finite elements which have derivative degrees of freedom are discussed. The first is Adini type elements which have derivative degrees of freedom on the corner nodes. The second is semiloof type elements which have normal derivative degree of freedom on the midpoint of the element edge. For the Adini element which has a rectangular shape, we propose a high-order mapping method to adapt to the structured quadrilateral shape. For the semiloof elements, we discuss the conforming and nonconforming cases. The discretization complexity of the semiloof element is lower than the Adini element. Numerical examples demonstrate the excellent performance of this approach.