A symmetric and coercive finite volume scheme preserving the discrete maximum principle for anisotropic diffusion equations on star-shaped polygonal meshes

A nonlinear vertex -centered finite volume scheme that preserves the discrete maximum principle (DMP) is proposed for anisotropic diffusion equations on polygonal meshes. The new scheme is constructed from a vertex -centered linearity -preserving scheme combined with a nonlinear correction technique. The key ingredient is the introduction of the correction coefficient and the limiter. Particularly, the choice of the limiter is discussed and two new and effective limiters are proposed. Different from most existing nonlinear DMP-preserving schemes, the new nonlinear scheme does not rely on the nonlinear combination of linear flux approximations and requires no interpolation of the auxiliary unknowns. More important is that, it can be proved theoretically that the new scheme is symmetric and coercive on general polygonal meshes with star -shaped cells. Moreover, other theoretical results such as the DMP property and stability of the correction scheme are also presented. Numerical experiments demonstrate the accuracy, efficiency, and DMP property of the new scheme on general polygonal meshes with heterogeneous and anisotropic diffusion tensors.