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.

翻译：本文针对线性与半线性抛物型方程的时间离散问题，研究了指数中点法的后验误差估计。采用由节点值的连续分段线性插值定义的指数中点逼近方法时，可获得次优阶估计。基于整函数性质，我们引入指数中点法的连续分段二次时间重构，推导出最优阶估计，且误差界仅依赖于离散化参数、问题数据及整函数逼近。通过多个数值算例验证了理论结果。