On the convergence of a combined finite volume-finite element method for nonlinear convection-diffusion problems. (English) Zbl 0869.65057

The authors consider a combined finite volume-finite element method for scalar conservation laws with small diffusion, and give a convergence analysis in the case of triangulations of the weakly acute type. The nonlinear convective terms are approximated by a monotone finite volume method on a mesh dual to a triangular grid. For the discretization of the diffusion term, a conforming finite element method with piecewise linear elements is used. With respect to time, the discrete problem is a linearized backward Euler (semi-implicit) scheme.
The convergence proof is carried out with the aid of the discrete maximum principle, a priori estimates, and compactness arguments based on the temporal Fourier transform. With the paper in hand, the theoretical analysis for an efficient numerical scheme for approximating boundary layers as well as slightly smeared shock waves arising from convection dominated convection-diffusion problems such as the dissipative Burgers equation is provided.


65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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