Some remarks on the convergence of approximate solutions of nonlinear evolution equations in Hilbert spaces

Abstract:

Let $\partial \Phi$ be the subdifferential of some lower semicontinuous convex function $\Phi$ of a real Hilbert space H, $f \in {L^2}(0,T;H)$ and ${u_n}$ a continouous piecewise linear approximate solution of $du/dt + \partial \Phi (u) \ni f$, obtained by an implicit scheme. If ${u_0} \in \operatorname {Dom} (\Phi )$, then $d{u_n}/dt$ converges to $du/dt$ in ${L^2}(0,T;H)$. Moreover, if ${u_0} \in \overline {\operatorname {Dom} (\partial \Phi )}$, we construct a step function ${\eta _n}(t)$ approximating t such that ${\lim _{n \to + \infty }}\smallint _0^T{\eta _n}|d{u_n}/dt - du/dt{|^2}\;dt = 0$. When $\Phi$ is inf-compact and when the sequence of approximation of f is weakly convergent to f, then ${u_n}$ converges to u in $C([0,T];H)$ and ${\eta _n}d{u_n}/dt$ is weakly convergent to $tdu/dt$.