A posteriori error estimates for fully discrete finite difference method for linear parabolic equations

In this paper, we study a posteriori error estimates for one-dimensional and two-dimensional linear parabolic equations. The backward Euler method and the Crank–Nicolson method for the time discretization are used, and the second-order finite difference method is employed for the space discretization. Based on linear interpolation and interpolation estimate, a posteriori error estimators corresponding to space discretization are derived. For the backward Euler method and the Crank–Nicolson method, the errors due to time discretization are obtained by exploring linear continuous approximation and two different continuous, piecewise quadratic time reconstructions, respectively. As a consequence, the upper and lower bounds of a posteriori error estimates for the fully discrete finite difference methods are derived, and these error bounds depend only on the discretization parameters and the data of the model problems. Numerical experiments are presented to illustrate our theoretical results.