  Motivated by a wide range of real-world problems whose solutions exhibit
boundary and interior layers, the numerical analysis of discretizations of
singularly perturbed differential equations is an established sub-discipline
within the study of the numerical approximation of solutions to differential
equations. Consequently, much is known about how to accurately and stably
discretize such equations on \textit{a priori} adapted meshes, in order to
properly resolve the layer structure present in their continuum solutions.
However, despite being a key step in the numerical simulation process, much
less is known about the efficient and accurate solution of the linear systems
of equations corresponding to these discretizations.
  In this paper, we discuss problems associated with the application of direct
solvers to these discretizations, and we propose a preconditioning strategy
that is tuned to the matrix structure induced by using layer-adapted meshes for
convection-diffusion equations, proving a strong condition-number bound on the
preconditioned system in one spatial dimension, and a weaker bound in two
spatial dimensions. Numerical results confirm the efficiency of the resulting
preconditioners in one and two dimensions, with time-to-solution of less than
one second for representative problems on $1024\times 1024$ meshes and up to
$40\times$ speedup over standard sparse direct solvers.
