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

Author

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  • Allaberen Ashyralyev
  • Pavel E. Sobolevskiĭ

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.

Suggested Citation

  • Allaberen Ashyralyev & Pavel E. Sobolevskiĭ, 2006. "Well-posedness of the difference schemes of the high order of accuracy for elliptic equations," Discrete Dynamics in Nature and Society, Hindawi, vol. 2006, pages 1-12, March.
    • Handle: RePEc:hin:jnddns:075153
    DOI: 10.1155/DDNS/2006/75153
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