  We consider discrete Schr\"odinger operators with aperiodic potentials given
by a Sturmian word, which is a natural generalisation of the
Fibonacci-Hamiltonian. We introduce the finite section method, which is often
used to solve operator equations approximately, and apply it first to periodic
Schr\"odinger operators. It turns out that the applicability of the method is
always guaranteed for integer-valued potentials. By using periodic
approximations, we find a necessary and sufficient condition for the
applicability of the finite section method for aperiodic Schr\"odinger
operators and a numerical method to check it.
