Well-posedness of the difference schemes of the high order of accuracy for elliptic equations. (English) Zbl 1102.65085
Summary: It is well known the differential equation \(-u''(t)+Au(t)=f(t)\)\((-\infty < t < \infty)\) in a general Banach space \(E\) with the positive operator \(A\) is ill-posed in the Banach space \(C(E)=C((-\infty,\infty),E)\) of the bounded continuous functions \(\varphi(t)\) defined on the whole real line with norm \(\|\varphi\|_{C (E)}=\sup_{-\infty < t < \infty}\|\varphi(t)\|_{E}\). In the present paper we consider the high order of accuracy two-step difference schemes generated by an exact difference scheme or by Taylor’s decomposition on three points for the approximate solutions of this differential equation. The well-posedness of these difference schemes in the difference analogy of the smooth functions is obtained. The exact almost coercive inequality for solutions in \(C(\tau,E)\) of these difference schemes is established.
MSC:
| 65L12 | Finite difference and finite volume methods for ordinary differential equations |
| 34G10 | Linear differential equations in abstract spaces |
| 65J10 | Numerical solutions to equations with linear operators |
| 65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |
| 35J40 | Boundary value problems for higher-order elliptic equations |
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