A combined finite volume–finite element scheme for the discretization of strongly nonlinear convection–diffusion–reaction problems on nonmatching grids

We propose and analyze a numerical scheme for nonlinear degenerate parabolic convection-diffusion-reaction equations in two or three space dimensions. We discretize the time evolution, convection, reaction, and sources terms on a given grid, which can be nonmatching and can contain nonconvex elements, by means of the cell-centered finite volume method. To discretize the diffusion term, we construct a conforming simplicial mesh with the vertices given by the original grid and use the finite element method. In this way, the scheme is fully consistent and the discrete solution is naturally continuous across the interfaces between the subdomains with nonmatching grids, without introducing any supplementary equations and unknowns or using any interpolation of the discrete solutions at the interfaces. We allow for general inhomogeneous and anisotropic diffusion-dispersion tensors and use the local Peclet upstream weighting in order to only add the minimal numerical diffusion necessary to avoid spurious oscillations in the convection-dominated case. The scheme is robust, efficient, locally conservative, and satisfies the discrete maximum principle under some conditions on the mesh and the diffusion tensor. We prove its convergence using a priori estimates and the Kolmogorov relative compactness theorem and illustrate its behavior on a numerical experiment.