A posteriori error estimates for the exponential midpoint method for linear and semilinear parabolic equations

In this paper, the a posteriori error estimates of the exponential midpoint method for time discretization are established for linear and semilinear parabolic equations. Using the exponential midpoint approximation defined by a continuous and piecewise linear interpolation of nodal values yields 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 optimal order estimates; the error bounds solely depend 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.