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Zbl 1166.65023
Ashyralyev, Allaberen
Well-posedness of the difference schemes for elliptic equations in $C_\tau^{\beta,\gamma}(E)$ spaces. (English)
[J] Appl. Math. Lett. 22, No. 3, 390-395 (2009). ISSN 0893-9659

The paper deals with a second order of accuracy difference scheme for the periodic (nonlocal) boundary value problem $$-v''(t)+Av(t)=f(t), \; t \in (0,1),$$ $$v(0)=v(1), \; v^{\prime}(0)=v^{\prime}(1)$$ with a strongly positive operator coefficient $A$ in a Banach space. A series of coercivity inequalities in difference analogues of various Hölder norms is obtained and the well-posedness of the difference scheme in $C_{\tau}^{\beta, \gamma}(E)$ spaces is proved. An example of an elliptic $2m$-order multidimensional partial differential equation is considered.
[Iwan Gawriljuk (Eisenach)]

MSC 2000:: *65J10 Equations with linear operators (numerical methods)
65N06 Finite difference methods (BVP of PDE)
65N12 Stability and convergence of numerical methods (BVP of PDE)
34G10 Linear ODE in abstract spaces
35J40 Higher order elliptic equations, boundary value problems

Keywords: elliptic differential equation; operator coefficient; Banach space; strongly positive operator; periodic boundary conditions; difference schemes; well-posedness; abstract elliptic problem

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