On the convergence of a combined finite volume-finite element method for nonlinear convection-diffusion problems. Explicit schemes

Abstract

This article is a continuation of the work [M. Feistauer et al., Num Methods PDEs 13 (1997), 163-190] devoted to the convergence analysis of an efficient numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume mesh dual to a triangular grid, whereas the diffusion term is discretized by piecewise linear conforming triangular elements. In the previous article [1] the convergence of a semi-implicit scheme was established. Here we are concerned with the analysis of fully explicit schemes. Under the assumption that the triangulations are of weakly acute type, with the aid of the discrete maximum principle, a priori estimates and some compactness arguments based on the use of the Fourier transform with respect to time, the convergence of the approximate solutions to the exact solution is proved, provided that the mesh size tends to zero.

L2-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

“…Discretizing this equation by the combined FE-FV scheme described above, with a rather general numerical flux adapted to the nonlinearity, and with a semi-implicit Euler method as time discretization, they derived L 2 (H 1 )-and L ∞ (L 2 )-error estimates. References [5,24,25,28] present results analogous to those in [2,18], but for a combined FE-FV method involving piecewise linear conforming finite elements and dual finite volumes (triangular finite volumes in the case of [5]). Similar L 2 (H 1 )-and L ∞ (L 2 )-error estimates as in [18] are shown in [27,50], but with respect to various discontinuous Galerkin schemes.…”

A domain decomposition method for two-phase transport model in the cathode of a polymer electrolyte fuel cell

“…In order to combine all the power of both finite element method and upwind scheme together, we introduce a combined finite element-upwind finite volume method [12,13,11] for the PEFC model in this section, where a finite volume based finite-difference upwind scheme is adopted to specifically deal with dominant convection term only, meanwhile, all the other terms are still discretized by finite element method. By doing this, we are able to take into account the irregular domain and natural boundary condition as well as dominant convection term without tuning any stabilization parameters.…”

Robust Numerical Methods for Singularly Perturbed Differential Equations

“…This approach has been extended to nonstationary nonlinear convection-diffusion and compressible flow problems in [FFLM95,FFLM97,FSS99]. Thus these methods are sometimes called combined finite volume-finite element methods.…”