  In this paper, for a new Stekloff eigenvalue problem which is non-selfadjoint
and not $H^1$-elliptic, we establish and analyze two kinds of two-grid
discretization scheme and a local finite element scheme. We present the error
estimates of approximations of two-grid discretizations. We also prove a local
error estimate which is suitable for the case that the local refined region
contains singular points lying on the boundary of domain. Numerical experiments
are reported finally to show the efficiency of our schemes.
