  In the first part of the paper, we study the discontinuous Galerkin (DG) and
$C^0$ interior penalty ($C^0$-IP) finite element approximation of the periodic
strong solution to the fully nonlinear second-order
Hamilton--Jacobi--Bellman--Isaacs (HJBI) equation with coefficients satisfying
the Cordes condition. We prove well-posedness and perform abstract a posteriori
and a priori analyses which apply to a wide family of numerical schemes. These
periodic problems arise as the corrector problems in the homogenization of HJBI
equations. The second part of the paper focuses on the numerical approximation
to the effective Hamiltonian of ergodic HJBI operators via DG/$C^0$-IP finite
element approximations to approximate corrector problems. Finally, we provide
numerical experiments demonstrating the performance of the numerical schemes.
