Combined Nonconforming/Mixed-hybrid Finite Element-Finite Volume Scheme for Degenerate Parabolic Problems

Summary

We propose and analyze an efficient numerical scheme for nonlinear degenerate parabolic convection-reaction-diffusion equations. We discretize the diffusion term, which generally involves a full matrix diffusion tensor, by means of piecewise linear nonconforming (Crouzeix-Raviart) finite elements over a triangulation of the space domain, or using the stiffness matrix of the hybridization of the lowest order Raviart-Thomas mixed finite element method. The other terms are discretized by means of a finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original triangulation. Checking the local Peclet number, we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the velocity dominated case. Under the regularity condition for the triangulation, using a priori estimates and Kolmogorov’s relative compactness theorem, the convergence of the scheme is proved.