In [16], Stynes and O'Riordan(91) introduced a local exponentially fitted finite element (FE) scheme for a singularly perturbed two-point boundary value problem without turning-point. An $ε$-uniform $h^{1/2}$-order accuracy was obtain for the $ε$-weighted energy norm. And this uniform order is known as an optimal one for global exponentially fitted FE schemes (see [6, 7, 12]).

In present paper, this scheme is used to a parabolic singularly perturbed problem. After some subtle analysis, a uniformly in $ε$ convergent order $h|{\rm ln} h|^{1/2}+ τ$ is achieved ($h$ is the space step and $τ$ is the time step), which sharpens the results in present literature. Furthermore, it implies that the accuracy order in [16] is actually $h|{\rm ln} h|^{1/2}$ rather than $h^{1/2}$.