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.

中文翻译：

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离散 Gagliardo-Nirenberg 不等式及其在具有焦耳效应项的对流扩散方程的有限体积近似中的应用

针对在由矩形单元组成的二维结构化网格上定义的分段常数函数，建立了离散二阶 Gagliardo-Nirenberg 不等式。与连续框架中一样，这种离散 Gagliardo-Nirenberg 不等式可以通过离散 Hessian 矩阵的 $L^2$ 范数乘以 $L^\ 来控制数值解离散梯度的 $L^4$ 范数。五十美元标准。这一结果对于有限体积法的收敛分析至关重要，该方法用于近似对流扩散方程，该方程涉及每个方向均匀网格上的焦耳效应项。收敛证明依赖于紧致性论证和数据较小假设下的先验估计，这在连续框架中也是必不可少的。