Analytical method for finding the nullspace of stiffness matrix in the finite element method

The stiffness matrix in a vector finite element method has a nullspace whose size grows linearly with the matrix size. Finding it numerically can be computationally expensive. However, the nullspace is important in many applications from solving the low-frequency breakdown to developing fast solvers. In this paper, we find an analytical method to generate the nullspace of the stiffness matrix without any need for computation. The method is valid regardless of the element shape used for discretization, the mesh configuration, and the basis-function order employed to expand the vector fields. Numerical experiments have validated the proposed analytical method.