DDFV scheme for nonlinear parabolic reaction-diffusion problems on general meshes

This paper focuses on the nonlinear anisotropic parabolic model of the form $\partial_{t}C(u)-\operatorname{div}(\Lambda \nabla u)+ R(u)=f$, where $C$, $R$, $f$, and $\Lambda$ are respectively: two nonlinear functions, a source term and an anisotropic tensor diffusion.  For space discretization, various types of the Discrete Duality Finite Volume (DDFV) scheme are elaborated leading to positive definite stiffness matrices for the diffusion term.  A general mesh is used and hard anisotropic tensor with discontinuous effects is considered.  An implicit time scheme is developed as well as the Newton–Raphson method to solve the resulting nonlinear system.  An iterative incremental approach is elaborated handling the effects of anisotropy, discontinuity and non-linearity.  The performance of the presented direct and indirect DDFV schemes for different meshes has been demonstrated by various numerical tests.  A super-convergence in the discrete $L^{2}$ and $H^{1}$-norms is also demonstrated.