Error analysis of the finite element approximations for hyperbolic partial integro-differential equations (Q2764267)

This paper discussed semi-discrete finite element approximation schemes of nonlinear hyperbolic partial integro-differential equations. The author discussed the superconvergence order in \( H'\) and optimal maximum norms in \(L^{\infty}\) and \(W^{1,\infty}\): NEWLINENEWLINENEWLINELet \(u, U\) are solution of a nonlinear hyperbolic partial integro-differential equations,and \( u \in W^{k+1,\infty}(\Omega) \cap L^2 (W^{k+1,\infty} (\Omega))\), then estimation of the optimal maximum norms in \(L^{\infty}\) and \(W^{1,\infty}\) are NEWLINE\[NEWLINE \|u-U\|_{s,\infty} \leq C h^{k+1-s}|\ln h |^{\overline{k} (1-s)}, \quad s = 0,1, NEWLINE\]NEWLINE where \( \overline{k} = \begin{cases} 1 , &k=1 \\ 0,&k \geq 2\end{cases}\).