 \\\le & {} \frac{\tau _m^0}{0} \int _\varOmega I^m_h \tilde{\rho }^m | \nabla e_h^{m}|^0 + C \tau _m^0 \epsilon \Vert u^m \Vert _{H^0(\varGamma (t_m))}^0. \end{aligned}$$\end{document}$$We deduce from ([00](#Equ00){ref-type=""}), Lemma [0](#FPar0){ref-type="sec"} a), Lemma [0](#FPar0){ref-type="sec"} and ([00](#Equ00){ref-type=""}) that$$\documentclass[00pt]{minimal}                 \usepackage{amsmath}                 \usepackage{wasysym}                  \usepackage{amsfonts}                  \usepackage{amssymb}                  \usepackage{amsbsy}                 \usepackage{mathrsfs}                 \usepackage{upgreek}                 \setlength{\oddsidemargin}{-00pt}                 \begin{document}$$\begin{aligned}&| \langle S^m_0,e_h^{m}\rangle | \le C \int _{t_{m-0}}^{t_m} \int _\varOmega \left[ | \