For stationary diffusion-type equations, we study the convergence of grid inhomogeneous boundary-value problems of a version of the mimetic finite difference (MFD) technique in which grid scalars are defined inside grid cells and grid vectors are specified by their local normal coordinates on the plane faces of grid cells. Grid equations and boundary conditions are formulated in operator form using consistent grid analogs of invariant first-order differential operators and of boundary operators. Convergence is studied on the basis of the theory of operator difference schemes; i.e., a priori estimates for the norm of the solution error in terms of the norm of the approximation error are obtained that guarantee convergence of the first order under inhomogeneous boundary conditions of the first, second, and third kind in a domain with a curvilinear boundary. Grid analogs of embedding inequalities and approximation relations obtained earlier are used.

中文翻译：

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网格单元和面上定义的函数的网格边值问题的收敛性

对于平稳扩散型方程，我们研究了一种模拟有限差分（MFD）技术的网格非均质边值问题的收敛性，该模型在网格单元内定义网格标量，并通过其局部法向坐标指定网格矢量。网格单元的平面。网格方程和边界条件使用不变的一阶微分算子和边界算子的一致网格类似物以算子形式表示。基于算子差分方案的理论研究收敛性；即先验获得关于近似误差范数的解误差范数的估计，以确保在具有曲线边界的域中第一，第二和第三类的非均匀边界条件下一阶收敛。使用较早获得的嵌入不等式和近似关系的网格模拟。