Well-posedness of the difference schemes of the high order of accuracy for elliptic equations

Allaberen Ashyralyev and Pavel E. Sobolevskiĭ

Discrete Dynamics in Nature and Society, 2006, vol. 2006, 1-12

Abstract:

It is well known the differential equation − u ″ ( t ) + A u ( t ) = f ( t ) ( − ∞ < t < ∞ ) in a general Banach space E with the positive operator A is ill-posed in the Banach space C ( E ) = C ( ( − ∞ , ∞ ) , E ) of the bounded continuous functions ϕ ( t ) defined on the whole real line with norm ‖ ϕ ‖ C ( E ) = sup ⁡ − ∞ < t < ∞ ‖ ϕ ( t ) ‖ E . In the present paper we consider the high order of accuracytwo-step difference schemes generated by an exact differencescheme or by Taylor's decomposition on three points for theapproximate solutions of this differential equation. Thewell-posedness of these difference schemes in the differenceanalogy of the smooth functions is obtained. The exact almostcoercive inequality for solutions in C ( τ , E ) of these difference schemes is established.

Date: 2006
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnddns:075153

DOI: 10.1155/DDNS/2006/75153

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