\(L_ \infty\)-estimates of quasi-optimal order for Galerkin methods to \(N=2,3\) dimensional second order hyperbolic differential equations (Q1323214)

The following model problem is considered: \(u_{tt} - \Delta u = f\) in \(\Omega \times (0,T)\), \(u = 0\) on \(\partial \Omega \times (0,T)\), \(u(0) = u_0\), \(u_t(0) = u_1\) in \(\Omega\). The aim of this paper is to give error estimates of the type \[ |e|_{L_\infty (L_\infty)} \leq ch^{m - N + 1} \{|u |_{L_\infty (W^{m - N + 1}_\infty)} + |u |_{L_2 (W^m_\infty)}\} \] for \(N = 2, 3\), where \(u_h = u_h(t)\) is the standard Galerkin approximation and \(e = u - u_h\) is the error.