Basis Functions With Divergence Constraints for the Finite Element Method

Basis functions for solving partial differential equations of vector fields using the finite element method are presented. The basis functions are a combination of cubic Hermite splines and second-order Lagrange interpolation polynomials and allow the divergence to be set as a constraint. The basis functions are tested on 3-D resonant cavities and there are no spurious modes. There is good agreement with analytical solutions in cases where they exist and with calculations using edge elements in other cases. The method is extended to solve problems with singularities at edges and corners. For perfect conductors, this includes mesh refinement in the neighborhood of the edge or corner. For dielectrics, constraints are derived so that the flux of the field is zero through a closed surface that contains the edge or corner. The method is used to solve problems using the electric and magnetic field formulations. Through a change of variables, the method is applied to problems in cylindrical coordinates.