Difference Schemes on Uniform Grids Performed by General Discrete Operators

Our aim is to set the foundations of a discrete vectorial calculus on uniform n-dimensional grids, that can be easily re-formulated on general irregular grids. As the key tool we first introduce the notion of tangent space to any grid node. Then we define the concepts of vector field, field of matrices and inner products on the space of grid functions and on the space of vector fields, mimicking the continuous setting. This allows us to obtain the discrete analogous of the basic first order differential operators, gradient and divergence, whose composition define the fundamental second order difference operator. As an application, we show that all difference schemes, with constant coefficients, for first and second order differential operators with constant coefficients can be seen as difference operators of the form −div (A∇u)+〈b,∇u〉+q u for suitable choices of q, b and A. In addition, we characterize special properties of the difference scheme, such as consistency, symmetry and positivity in terms of q, b and A.