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Error estimates over infinite intervals of some discretizations of evolution equations. (English) Zbl 0573.65038
The author considers the initial value problem in a reflexive B-space $$V: du/dt+F(t,u)=0$$, $$u(0)=u_ 0\in V$$, $$t>0$$. The operator $$F: V\to V'$$ is not necessarily bounded. Four cases are investigated: F monotone, F strongly monotone, F conservative, F non-monotone (the case (F(t,u)- F(t,v), u-v)$$\leq -\rho_ 0\| u-v\|^ 2$$, $$\forall u,v\in V$$ with $$\rho_ 0>0)$$. The (implicit) $$\vartheta$$-method is applied: $$v(t-k)- v(t)+kF(t+(1-\vartheta)k,\vartheta v(t)+(1-\vartheta)v(t+k))=0$$ with $$0\leq \vartheta \leq 1$$. It is assumed that this nonlinear equation has a unique solution in V.
In each of the discussed cases estimates for the error $$e(t)=u(t)-v(t)$$ are given in terms of e(0) and of the truncation error $R_{\vartheta}(t,u)=F(t+(1-\vartheta)k,\vartheta u(t)+(1- \vartheta)u(t+k))-k^{-1}\int^{t+k}_{t}F(s,u(s))ds.$ Many remarks on the behaviour of solutions are joint, also the so called ’forced solution’ (slowly increasing) in the non-monotone case is discussed. One of the sections of the paper is devoted to estimates of the truncation error. The last section discusses the so called boundary value techniques, in the context of the forced solution for the non-monotone case. The method presented here is composed of the repeated midpoint formula and the backward Euler on the last step only.
Reviewer: K.Moszyński

MSC:
 65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx) 65L05 Numerical methods for initial value problems 34G20 Nonlinear differential equations in abstract spaces
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