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Averaging of difference schemes. (English. Russian original) Zbl 0639.65052

Math. USSR, Sb. 57, 351-369 (1987); translation from Mat. Sb., Nov. Ser. 126(171), No. 3, 338-357 (1986).
The author considers the family of difference operators \(A_{\epsilon}u=\epsilon^{-2}(\sum_{y}p^{\epsilon}(x,y)u(y)-u(x))\) on the uniform grid of the mesh \(\epsilon\) ; x,y\(\in Q\), a bounded domain in \(R^ n\). Here \(p^{\epsilon}(x,y)=p^{\epsilon}(y,x)\geq 0\) (x\(\neq y)\); \(| p^{\epsilon}(x,x)| <C_ 0\); \(\sum_{y}p^{\epsilon}(x,y)=1\); \(p^{\epsilon}(x,y)=1\); \(p^{\epsilon}(x,y)=0\) if \(| x-y| >c,\epsilon\); \(\sum_{i}p^{\epsilon}(x,x+\epsilon e_ i)\xi_ i^ 2\geq \delta | \xi |^ 2.\) It is shown that there exist some sequence \(\epsilon\) ’\(\to 0\) and some elliptic operator A such that \(A_{\epsilon '}\) approximate A in any generalize sence. Some applications are given.
Reviewer: V.V.Vasil’ev

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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