Removal of spurious DC modes in edge element solutions for modeling three-dimensional resonators

When using edge element basis functions for the solution of eigenmodes of the vector wave equation, "dc spurious modes" are introduced. The eigenvalues of these modes are zero and their corresponding eigenvectors are in the null space of the curl operator. These modes arise due to the irrotational vector space spanned by the edge element basis functions and lead to nonzero divergence of the electric flux. We introduce a novel method to eliminate the occurrence of such solutions using "divergence-free" constraint equations. The constraint equations are imposed efficiently by tree-cotree partitioning of the finite-element mesh and does not require any basis functions other than the edge elements. The constraint equations can be directly incorporated into any Krylov-subspace-based eigenvalue solver, such as the Lanczos/Arnoldi algorithm used widely for the solution of generalized sparse eigenvalue problems.