The semidiscrete finite volume element method for nonlinear convection-diffusion problem (Q544053)

A nonlinear convection-diffusion problem in \(\mathbb{R}^2\) is solved numerically with a semi-discrete finite volume method. The basic idea is to approximate the discrete fluxes of the partial differential equation using a finite element procedure based on control volumes. Results on the stability of the semi-discrete method as well as the existence and uniqueness of the solution provided by the scheme are provided. In the second part of the paper, a two-grid formulation of the (semi-discrete) finite volume method is introduced to solve the nonlinear parabolic problem. In this approach, two regular triangulations of the domain with mess size \(H\) and \(h \ll H\) are considered. The nonlinear problem is solved directly on the coarse grid and then the provided solution is used to construct the fine grid solution. The original problem becomes linear in this second stage of the procedure, and thus is much simpler to solve. \(L^2\)-norm and \(H^1\)-norm error estimates are also provided. A numerical example is included to illustrate the theoretical analysis.