  This study proposes a class of augmented subspace schemes for the weak
Galerkin (WG) finite element method used to solve eigenvalue problems. The
augmented subspace is built with the conforming linear finite element space
defined on the coarse mesh and the eigenfunction approximations in the WG
finite element space defined on the fine mesh. Based on this augmented
subspace, solving the eigenvalue problem in the fine WG finite element space
can be reduced to the solution of the linear boundary value problem in the same
WG finite element space and a low dimensional eigenvalue problem in the
augmented subspace. The proposed augmented subspace techniques have the second
order convergence rate with respect to the coarse mesh size, as demonstrated by
the accompanying error estimates. Finally, a few numerical examples are
provided to validate the proposed numerical techniques.
