In this contribution we analyze the large time behavior of a family of nonlinear finite volume schemes for anisotropic convection-diffusion equations set in a bounded bidimensional domain and endowed with either Dirichlet and / or no-flux boundary conditions. We show that solutions to the two-point flux approximation (TPFA) and discrete duality finite volume (DDFV) schemes under consideration converge exponentially fast toward their steady state. The analysis relies on discrete entropy estimates and discrete functional inequalities. As a biproduct of our analysis, we establish new discrete Poincare-Wirtinger, Beckner and logarithmic Sobolev inequalities. Our theoretical results are illustrated by numerical simulations.

中文翻译：

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对流-扩散方程的非线性有限体积方案的长时间特性

在这个贡献中，我们分析了在有界二维域中设置的各向异性对流扩散方程的一系列非线性有限体积方案的大时间行为，并赋予了狄利克雷和/或无通量边界条件。我们表明，所考虑的两点通量近似 (TPFA) 和离散对偶有限体积 (DDFV) 方案的解决方案以指数方式快速收敛到稳定状态。该分析依赖于离散熵估计和离散函数不等式。作为我们分析的副产品，我们建立了新的离散 Poincare-Wirtinger、Beckner 和对数 Sobolev 不等式。我们的理论结果通过数值模拟来说明。