Spectral aspects of Nitsche's method on nonconforming meshes

Nitsche's method is an effective framework for the solution of problems involving embedded domains. Weak enforcement of Dirichlet boundary and interface conditions engenders additional active degrees of freedom compared to the corresponding kinematically admissible formulation, and hence additional solutions in eigenvalue problems. The original and added eigenpairs are designated proper and complementary, respectively. The number of complementary pairs equals the number of degrees of freedom that would be constrained in the kinematically admissible formulation. We investigate the number of complementary pairs that arise in representative nonconforming configurations of bilinear quadrilaterals. Algebraic elimination of the added degrees of freedom from the Nitsche formulation yields a formulation with several advantageous features. Practical procedures for solving eigenvalue problems based on reduced methods are proposed.