The L2-error estimates are established for the continuous time Faedo-Galerkin approximation to solutions of a linear parabolic initial boundary value problem that has elliptic part of order 2m. Properties of analytic semigroups are used to obtain these estimates directly from the LZ-estimates for the corresponding steady state elliptic problem under hypotheses only on the data in the problem (initial condition, elliptic operator). 1. Introduction.We obtain estimates for the error resulting from a continuous time Faedo-Galerkin approximation of the linear parabolic boundary value problem (1.1) u'(t) + Au(t) 0," > O, u(O) Uo, where A is a realization in LZ(G) of an elliptic partial differential operator of order 2m. These estimates are best possible: the rate ofconvergence is the same as that for the Galerkin approximation of the corresponding (variational) elliptic steady state problem whose exact solution is the initial condition, u0. These estimates for the rate of convergence are well known; our contribution here is that they are obtained from hypotheses on the data in the problem--the regularity properties of the elliptic operator, an approximation assumption on the rate of convergence in the corresponding elliptic problem and the initial condition --and without the usual ad-hoc assumptions on the solution u(. of the problem. The proofs depend on the existence-regularity theory for the evolution problem. See (1), (2), (12) for related results. An exposition of the well-known results for the steady state problem is given in 2, where we briefly discuss the approximation of solutions and the interpola- tion of various estimates associated with these regular elliptic boundary value problems. Section 3 begins with a description of the regularity properties of the solution of the abstract Cauchy problem (1.1) where -A is the generator of an analytic semigroup ofcontractions. We use the notions ofinterpolation theory and fractional powers of the operator A to relate the growth of u(t) in various norms as 0 + to the (regularity of the) initial condition, u0. After these preliminaries, the