  In the framework of abstract linear inverse problems in infinitedimensional
Hilbert space we discuss generic convergence behaviours of approximate
solutions determined by means of general projection methods, namely outside the
standard assumptions of Petrov-Galerkin truncation schemes. This includes a
discussion of the mechanisms why the error or the residual generically fail to
vanish in norm, and the identification of practically plausible sufficient
conditions for such indicators to be small in some weaker sense. The
presentation is based on theoretical results together with a series of model
examples and numerical tests.
