Weak finite difference approximation of a divergence differential operator of arbitrary order (Q1975774)

A quasilinear divergence differential operator of order \(m\) \[ F[u(x)] = \sum_{|\alpha|= 0}^m \text{D}^{\alpha} \cdot f_{\alpha}(u)\tag{1} \] is considered where \(x = (x_1,\dots,x_n)\), \(u(x)\), \(f_{\alpha} (u)\) are preassigned smooth functions; \(\alpha = (\alpha_1,\dots,\alpha_n)\), \(|\alpha|= \alpha_1 + \dot s +\alpha_n\) and \[ \text{D}^{\alpha} \cdot f = \frac{\partial^{|\alpha|}}{\partial x_1^{ \alpha_1} \dots \partial x_n^{\alpha_n}}, \quad \text{D}^0 \cdot f = f. \] The author introduces the concept of weak approximation of the divergence differential operator by the difference operator. Necessary and sufficient conditions under which the fifference operator satisfies the concept of weak approximation are obtained. Some corollaries of the main theorem are presented.