  In this article we study a piecewise linear discretization schemes for
transfer operators (Perron-Frobenius operators) associated with interval maps.
We show how these can be used to provide rigorous {\bf pointwise}
approximations for invariant densities of Markov interval maps. We also derive
the order of convergence of the approximate invariant density to the real one
in the $L^{\infty}$-norm. The outcome of this paper complements rigorous
results on $L^1$ approximations of invariant densities \cite{KMY} and recent
results on the formulae of escape rates of open dynamical systems \cite{KL2}.
We implement our computations on two examples (one rigorous and one
non-rigorous) to illustrate the feasibility and efficiency of our schemes.
