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On some aspects of the discontinuous Galerkin finite element method for conservation laws. (English) Zbl 1013.65108
Summary: The paper is concerned with the numerical solution of nonlinear conservation laws and nonlinear convection-diffusion problems. We discuss two versions of this method: (a) Finite volume discontinuous Galerkin (FVDG) method, which is a generalization of the combined finite volume-finite element (FV-FE) method. Its advantage is the use of only one mesh (in contrast to the combined FV-FE schemes). However, it is of the first order only. (b) Further, the pure DGFE method of higher order is considered. In this case, a new limiting is developed to avoid spurious oscillations in the vicinity of shocks.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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