Difference operator approximations on nonstandard rectangular grid

Difference methods are widely used for the approximate solution of boundary value problems for partial differential equations. Grid approximations are most simply constructed when the computational domain is divided into rectangular cells. Typically, the grid nodes coincide with the vertices of the cells. In addition to such node-center approximations, grids with nodes at the centers of cells are also used. It is convenient to formulate boundary value problems in terms of invariant operators of vector (tensor) analysis, which are associated with corresponding grid analogs. In this work, analogs of the gradient and divergence operators are constructed on non-standard rectangular grids the nodes of which consist of both the vertices of the computational cells and their centers. The proposed approach is illustrated using approximations of a boundary value problem for a stationary two-dimensional convection–diffusion equation. The key features of constructing approximations for vector problems are discussed with a focus on applied problems of the mechanics of solids.