This work addresses the development and analysis of a second-order accurate finite volume scheme for parabolic equations with anisotropy on general simplicial meshes. The discretization involves only vertex unknowns without processing additional ones. The scheme construction makes use of a nonlinear transformation of the linear elliptic term. Two propositions are mainly presented for the approximation of the mobility function at the interfaces. The existence of positive solutions for the discrete system is guaranteed thanks to the proved a priori estimates. The energy dissipation of the scheme is moreover ensured. The convergence of the approach is established. Numerical tests are given to show the efficiency, accuracy and robustness of the proposed approach, with respect to the anisotropy, while a particular emphasis is set on the effects of the approximate mobility. They also confirm the obtained theoretical results, especially the decay of the free energy when time grows.

这项工作解决了一般单纯网格上具有各向异性的抛物线方程的二阶精确有限体积方案的开发和分析。离散化只涉及顶点未知数，而没有处理额外的顶点未知数。方案构造利用线性椭圆项的非线性变换。主要提出两个命题来逼近界面处的迁移率函数。由于已证明的先验估计，保证了离散系统的正解的存在。此外确保了该方案的能量耗散。建立方法的收敛性。给出了数值测试以显示所提出的方法的效率、准确性和鲁棒性，关于各向异性，同时特别强调近似流动性的影响。他们还证实了所获得的理论结果，尤其是随着时间的增长自由能的衰减。