  We study discrete Schr\"odinger operators $H$ with periodic potentials as
they are typically used to approximate aperiodic Schr\"odinger operators like
the Fibonacci Hamiltonian. We prove an efficient test for applicability of the
finite section method, a procedure that approximates $H$ by growing finite
square submatrices $H_n$. For integer-valued potentials, we show that the
finite section method is applicable as soon as $H$ is invertible. This
statement remains true for $\{0, \lambda\}$-valued potentials with fixed
rational $\lambda$ and period less than nine as well as for arbitrary
real-valued potentials of period two.
