  We adapt a symmetric interior penalty discontinuous Galerkin method using a
patch reconstructed approximation space to solve elliptic eigenvalue problems,
including both second and fourth order problems in 2D and 3D. It is a direct
extension of the method recently proposed to solve corresponding boundary value
problems, and the optimal error estimates of the approximation to
eigenfunctions and eigenvalues are instant consequences from existing results.
The method enjoys the advantage that it uses only one degree of freedom on each
element to achieve very high order accuracy, which is highly preferred for
eigenvalue problems as implied by Zhang's recent study [J. Sci. Comput. 65(2),
2015]. By numerical results, we illustrate that higher order methods can
provide much more reliable eigenvalues. To justify that our method is the right
one for eigenvalue problems, we show that the patch reconstructed approximation
space attains the same accuracy with fewer degrees of freedom than classical
discontinuous Galerkin methods. With the increasing of the polynomial order,
our method can even achieve a better performance than conforming finite element
methods, such methods are traditionally the methods of choice to solve problems
with high regularities.
