A Posteriori Error Estimates for Exponential Midpoint Integrator Finite Element Method for Parabolic Equations

Abstract

In this paper, a posteriori error estimates for the fully discrete approximation for linear and semilinear parabolic equations are established. The exponential midpoint method is used for the temporal discretization, and the spatial discretization takes the standard linear elements. By introducing a continuous piecewise quadratic reconstruction, the error of the temporal discretization is derived. The error of the spatial discretization is evaluated by the interpolation estimates. Using the energy technique, we derive residual‐based a posteriori lower and upper bounds for the fully discrete approximation. Our theoretical results are supported by extensive numerical experiments.