An initial boundary value problem of
convection-diffusion type for a singularly perturbed quasilinear
parabolic equation is considered on an interval. For this problem
we construct e-uniformly
convergent difference schemes (nonlinear iteration-free schemes
and their iterative variants) based on the
domain decomposition method, which allow us to
implement sequential and parallel computations on decomposition
subdomains. Such schemes are obtained by domain decomposition
applied to an e-uniformly
convergent nonlinear base scheme, which is
a classic difference approximation of the differential
problem on piecewise uniform meshes condensing in a boundary
layer. The decomposition schemes constructed in this paper
converge e-uniformly
at the rate of *O*(*N*^{ -1}
ln *N* + *N*_{0}^{-1})*N* and *N*_{0}
denote respectively the number
of mesh intervals in the space and time discretizations.