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

TL;DR

This paper develops a posteriori error estimates for the exponential midpoint method applied to linear and semilinear parabolic equations, improving the accuracy of error bounds through a novel quadratic time reconstruction.

Contribution

It introduces a continuous, piecewise quadratic time reconstruction to achieve optimal order error estimates for the exponential midpoint method.

Findings

Error bounds depend only on discretization parameters and data

The new method achieves optimal order estimates

Numerical examples confirm theoretical results

Abstract

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.