I have a vague understanding of the compatibility equations for linear elasticity. They appear to be necessary in obtaining a unique displacement field. However, why is it that, in my papers I've come across, the compatibility equations aren't solved for in the linear elastic numerical formulation? Are they not a governing set of equations for linear elasticity?
A reasonably complete discussion of compatibility can be found at https://en.wikipedia.org/wiki/Compatibility_(mechanics)
There are six strain-displacement relations and only three displacements in continuum mechanics. The compatibility conditions will be needed if you wish to compute displacements given strains. However, most numerical formulations of mechanics are displacement-driven and the need for those equations typically does not arise because one uses: displacement -> strain -> stress -> force.
For force-driven formulations, one has to make sure that the displacements are compatible. That is because the direction of computations goes as: force -> stress -> strain -> displacement.