Efficient constrained finite element solution for electromagnetic eigenvalue problems with lossy anisotropic materials

Use of divergence-free constraint equations to suppress the appearance of suprious modes in the vector FEM modeling of electromagnetic resonators is extended to the case of lossy resonators. Electric field based FEM formulation for lossy Maxwell eigenvalue problem results in a quadaratic eigenvalue problem which can be linearized and cast into a standard eigenvalue problem. When an iterative technique such as the Lanczos/Arnoldi method is used for the eigenvalue solution, non-physical modes which donot satisfy the divergence-free condition of the electric flux appear along with the physical eigenvector solutions. Appearance of such eigenmodes is eliminated by constraining the eigenvector solution to be divergence free in a weak sense. The constraint equation is imposed efficiently by devising a gradient projector operator based on the tree-cotree partitioning of the finite element mesh. Coupling of the constraint equation with ARPACK routines invoked with the shift-and-invert strategy resulting in the eigenvalue solution free of non-physical eigenvalues is demonstrated.