We propose a kernel compensation mimetic difference (MD) scheme to solve the grad-div eigenvalue problem. This method applies stencil-based MD operators to discretize the gradient, curl and divergence operators and load the discrete grad-div operator. We construct the curl-curl compensation operator with a proper boundary condition that is orthogonal to the discrete grad-div operator. A generalized identification method for spurious eigenvalues is presented to eliminate the spectral components of the compensation operator to achieve no-pollution. The resulting scheme offers several advantages, including high-order accuracy, enhanced computational efficiency with reduced memory usage, and excellent scalability for parallel computation. Numerical tests demonstrate that our approach not only converges at the expected rates but also performs satisfactorily in terms of speed.