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

Abstract

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