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A finite element discretization is proposed and analyzed for a linear parabolic problems with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Numer. Math., 79 (1998), pp. 175--202]. In this paper, we have used a finite element discretization, where interface triangles are assumed to be curved triangles instead of straight triangles as in classical finite element methods. Optimal order error estimates in L2 and H1 norms are shown to hold even if the regularity of the solution is low on the whole domain. While the continuous time Galerkin method is discussed for the spatially discrete scheme, the discontinuous Galerkin method is analyzed for the fully discrete scheme. The interfaces and boundaries of the domains are assumed to be smooth for our purpose. [ABSTRACT FROM AUTHOR]