Discrete Gagliardo–Nirenberg inequality and application to the finite volume approximation of a convection–diffusion equation with a Joule effect term

A discrete order-two Gagliardo–Nirenberg inequality is established for piecewise constant functions defined on a two-dimensional structured mesh composed of rectangular cells. As in the continuous framework, this discrete Gagliardo–Nirenberg inequality allows to control in particular the $L^4$ norm of the discrete gradient of the numerical solution by the $L^2$ norm of its discrete Hessian times its $L^\infty $ norm. This result is crucial for the convergence analysis of a finite volume method for the approximation of a convection–diffusion equation involving a Joule effect term on a uniform mesh in each direction. The convergence proof relies on compactness arguments and on a priori estimates under a smallness assumption on the data, which is essential also in the continuous framework.