  We consider 1D discrete Schr\"odinger operators with aperiodic potentials
given by a Sturmian word, which is a natural generalisation of the Fibonacci
Hamiltonian. Via a standard approximation by periodic potentials, we establish
Hausdorff convergence of the corresponding spectra for the Schr\"odinger
operators on the axis as well as for their compressions to the half-axis. Based
on the half-axis results, we study the finite section method, which is another
operator approximation, now by compressions to finite but growing intervals,
that is often used to solve operator equations approximately. We find that,
also for this purpose, the aperiodic case can be studied via its periodic
approximants. Our results on the finite section method of the aperiodic
operator are illustrated by confirming a result on the finite sections of the
special case of the Fibonacci Hamiltonian.
