Nonconforming finite element approximations of the Steklov eigenvalue problem

This paper deals with nonconforming finite element approximations of the Steklov eigenvalue problem. For a class of nonconforming finite elements, it is shown that the j-th approximate eigenpair converges to the j-th exact eigenpair and error estimates for eigenvalues and eigenfunctions are derived. Furthermore, it is proved that the j-th eigenvalue derived by the element gives lower bound of the j-th exact eigenvalue, whereas the nonconforming Crouzeix–Raviart element and the element provide lower bounds of the large eigenvalues. Numerical results are presented to confirm the considered theory.