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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.

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 |

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

34G20 | Nonlinear differential equations in abstract spaces |

### Keywords:

theta-method; stability; forced solution; stiffness; initial value problem; reflexive B-space; monotone; strongly monotone; conservative; non-monotone; truncation error### References:

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