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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.

MSC:

65J10 Numerical solutions to equations with linear operators
65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
34G10 Linear differential equations in abstract spaces
35J40 Boundary value problems for higher-order elliptic equations
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