Method for Analytically Finding the Nullspace of Stiffness Matrix for Both Zeroth-Order and Higher Order Curl-Conforming Vector Bases in Unstructured Meshes

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 problem to developing fast solvers. In this article, an analytical method to generate the nullspace of the stiffness matrix is developed, not only for the zeroth-order but also for any higher order vector basis function in arbitrary unstructured meshes. Using the mesh information, we are able to analytically construct the nullspace without any difficulty, thus avoiding solving an eigenvalue problem for finding the nullspace. The proposed analytical method has been applied to a variety of 2-D and 3-D irregular meshes for various orders of vector basis functions. Comparisons with the reference nullspace obtained from a brute-force eigenvalue solution have validated the proposed analytical method. The proposed work has also been applied to efficiently solve the low-frequency breakdown problem in a full-wave solver and perform fast layout parasitics extraction.