I'm working on some adaptive discontinuous Galerkin codes for time harmonic wave propagation, currently just Helmholtz, but will be branching out once I have a working prototype in this case.
There are some papers out there demonstrating that residual based a posteriori estimators combined with a proper marking strategy yield a provable error reduction provided the refinement satisfies some conditions.
Now I am not an expert on a posteriori estimation. However the conditions of these theorems for error reduction appear to rely strongly on the fact that the solutions satisfy the specified discrete weak forms exactly. Unfortunately a direct solve is only possible for me on a few levels of refinement, then the problem becomes quite large. (The initial mesh has to be fine enough to resolve the wave frequency to ensure the estimators are valid, so it's not necessarily the case that I'm beginning with only a few elements)
I can't however find any literature detailing the relationship between the error incurred in an inexact linear solve and its impact on the reliability of convergence theorems for adaptivity. Is there any out there I might be missing, or maybe I'm just asking for too much?