  A singularly perturbed parabolic problem of convection-diffusion type with
incompatible inflow boundary and initial conditions is examined. In the case of
constant coefficients, a set of singular functions are identified which match
certain incompatibilities in the data and also satisfy the associated
homogenous differential equation. When the convective coefficient only depends
on the time variable and the initial/boundary data is discontinuous, then a
mixed analytical/numerical approach is taken. In the case of variable
coefficients and the zero level of compatibility being satisfied (i.e.
continuous boundary/initial data), a numerical method is constructed whose
order of convergence is shown to depend on the next level of compatibility
being satisfied by the data. Numerical results are presented to support the
theoretical error bounds established for both of the approaches examined in the
paper.
