  In this paper, the a posteriori error estimates of the exponential midpoint
method for time discretization are studied for linear and semilinear parabolic
equations. Using the exponential midpoint approximation defined by a continuous
and piecewise linear interpolation of nodal values yields the suboptimal order
estimates. Based on the property of the entire function, we introduce a
continuous and piecewise quadratic time reconstruction of the exponential
midpoint method to derive the optimal order estimates, and the error bounds are
solely dependent on the discretization parameters, the data of the problem and
the approximation of the entire function. Several numerical examples are
implemented to illustrate the theoretical results.
