On the convergence of difference approximations to nonlinear contraction semigroups in Hilbert spaces

Abstract

Convergence properties of the difference schemes (S) \[ h − 1 ∑ j = 0 k α j u n + j + ∑ j = 0 k β j A u n + j = 0 , n ⩾ 0 , {h^{ - 1}}\sum \limits _{j = 0}^k {{\alpha _j}{u_{n + j}}} + \sum \limits _{j = 0}^k {{\beta _j}A{u_{n + j}}} = 0,\quad n \geqslant 0, \] , for evolution equations (E) \[ d u ( t ) d t + A u ( t ) = 0 , t ⩾ 0 ; u ( 0 ) = u 0 ∈ D ( A ) ¯ \frac {{du(t)}}{{dt}} + Au(t) = 0,\quad t \geqslant 0;\quad u(0) = {u_0} \in \overline {D(A)} \] are studied. Here A is a nonlinear, maximally monotone operator in a real Hilbert space. It is shown, in particular, that if the scheme (S) is consistent and stable for the test equation x ′ = λ x x\prime = \lambda x for λ ∈ C − K \lambda \in {\text {C}} - K , where K is a compact subset of the right half-plane, then (S) is convergent as h ↓ 0 h \downarrow 0 , with suitable initial values, for (E), on compact intervals [0, T]. Moreover, the convergence is uniform on the half-axis t ⩾ 0 t \geqslant 0 , if the solution u ( t ) u(t) tends strongly to a constant as t → ∞ t \to \infty . We also show that under weaker stability conditions one can construct conditionally convergent methods.