  We present a simple two-dimensional model of the indirect dissociative
recombination process. The model has one electronic and one nuclear degree of
freedom and it can be solved to high precision, without making any physically
motivated approximations, by employing the exterior complex scaling method
together with the finite-elements method and discrete variable representation.
The approach is applied to solve a model for dissociative recombination of
H$_2^+$ and the results serve as a benchmark to test validity of several
physical approximations commonly used in the computational modeling of
dissociative recombination for real molecular targets. The second, approximate,
set of calculations employs a combination of multi-channel quantum defect
theory and frame transformation into a basis of Siegert pseudostates. The cross
sections computed with the two methods are compared in detail for collision
energies from 0 to 2 eV.
