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

Abstract

We present 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. 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.…”

$$L^\infty $$ -Stability of IMEX-BDF2 Finite Volume Scheme for Convection-Diffusion Equation

“…The hypothesis (H2), necessary to establish error estimates, is classical for the vertex-centered finite volume scheme [12] or the combined finite volumefinite element scheme [9]. Obviously, the M-matrix property still holds for Delaunay triangulations (see [6], Sect.…”

Analysis of a combined barycentric finite volume—nonconforming finite element method for nonlinear convection-diffusion problems

“…Since we use nonconforming FEM and thus V h ⊂ V , the convergence analysis is more complex than in the conforming case investigated in [11]. Our further considerations will be based on results from [31] and [8], Section 8.9.…”

“…Since the complete viscous gas flow problem is rather complex, the theoretical analysis of the combined finite volume-finite element method has been carried out for the case of a simplified scalar nonlinear conservation law equation with a small dissipation which is the simplest prototype of the compressible Navier-Stokes equations. Papers [11], [13], [15] are concerned with the convergence and error estimates for the method using dual finite volumes over a triangular mesh combined with conforming piecewise linear triangular finite elements.…”