Difference schemes on uniform grids performed by general discrete operators
The paper is devoted to rather formal constructions of grid approximations on uniform grids for \(n\)-dimensional linear elliptic equations. As a key tool, difference analogues of gradient and divergence operators are used. The first and second order approximations are considered. Under the difference scheme, ``an expression of the form \(L_h(u)(x)\)'' (a difference formula) is understood which is defined mainly for finite functions on the uniform grid in \(\mathbb R^n\). Reviewer's remarks: Grid analogues of gradient and divergence operators and their applications were considered in many papers [for example, \textit{V. I. Lebedev}, Zh. Vychisl. Mat. Mat. Fiz. 4, 449--465, 649--659 (1964; Zbl 0158.34301)], but for the case of more involved boundary conditions it is known that grid analogues of the basic integral identity are more helpful. Some notions in the paper are not properly defined; for example, on page 345 the definition of ``\(r\)-consistent scheme'' makes use of \(O(h^r)\) estimate for a grid function without specification of its sense (it is well known that the choice of the norm is very important).
scientific article; zbMATH DE number 2093395