Optimal order constructive a priori error estimates for a full discrete approximation of the heat equation

In this paper, we consider constructive a priori error estimates for a full discrete numerical solution of parabolic initial boundary value problems. Our method is based on the finite element Galerkin method with an inter-polation in time that uses the fundamental solution for semidiscretization in space. Particularly, we present optimal order error estimates for the linear finite element in both space and time directions. These error estimates are sharper than the existing results in the sense of convergence order to exact solutions. Since the sharply constructive error estimates play an essential role in improving the efficiency of the verification costs, our results are expected to contribute to a new development of the numerical proof for parabolic problems. We also present some numerical examples which confirm that our estimates have the exactly the same order of convergence as the a posteriori errors.