In this paper, we propose error estimates that provide the explicit values of the error constants for the fully-discrete Petrov-Galerkin approximation of the linear heat equation in the \(L^2(J;H^1_0(\varOmega ))\) norm, the \(L^2(J;L^2(\varOmega ))\) norm, and the \(L^2(\varOmega )\) norm for a fixed time, where J is a time interval and \(\varOmega \) is a bounded domain. The error estimates for the approximation were studied to show the convergence in some norms to the weak solution of the linear heat equation. Furthermore, methods for computing the explicit values of the error constants for the estimates play an important role in computer-assisted existence proofs of solutions to semi-linear parabolic partial differential equations. However, computers cannot always derive explicit values using existing methods because they have to compute the rigorous value of the Euclidean norm of the matrices, whose sizes become larger as the approximation converges to the weak solution. In this paper, we solve this difficulty by proposing norm estimation independent of the values of the Euclidean norm.
