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Well-posedness of the difference schemes for elliptic equations in $C_\tau^{\beta,\gamma}(E)$ spaces. (English) Zbl 1166.65023
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.

65J10Equations with linear operators (numerical methods)
65N06Finite difference methods (BVP of PDE)
65N12Stability and convergence of numerical methods (BVP of PDE)
34G10Linear ODE in abstract spaces
35J40Higher order elliptic equations, boundary value problems
Full Text: DOI
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