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

Author: Olavi Nevanlinna
Journal: Math. Comp. 32 (1978), 321-334
MSC: Primary 47H15; Secondary 65J05
MathSciNet review: 0513203
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Abstract: Convergence properties of the difference schemes (S)

$\displaystyle {h^{ - 1}}\sum\limits_{j = 0}^k {{\alpha _j}{u_{n + j}}} + \sum\limits_{j = 0}^k {{\beta _j}A{u_{n + j}}} = 0,\quad n \geqslant 0,$

, for evolution equations (E)

$\displaystyle \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\prime = \lambda x$ for $ \lambda \in {\text{C}} - K$, where K is a compact subset of the right half-plane, then (S) is convergent as $ h \downarrow 0$, with suitable initial values, for (E), on compact intervals [0, T]. Moreover, the convergence is uniform on the half-axis $ t \geqslant 0$, if the solution $ u(t)$ tends strongly to a constant as $ t \to \infty $. We also show that under weaker stability conditions one can construct conditionally convergent methods.

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Article copyright: © Copyright 1978 American Mathematical Society