Analysis of the discontinuous Galerkin finite element method applied to a scalar nonlinear convection-diffusion equation. (English) Zbl 1426.35189

Hozman, Jiří; Dolejší, Vít

Chleboun, J. (ed.) et al., Programs and algorithms of numerical mathematics 14. Proceedings of the seminar, Dolní Maxov, Czech Republic, June 1–6, 2008. Prague: Academy of Sciences of the Czech Republic, Institute of Mathematics. 97-102 (2008).

Summary: We deal with a scalar nonstationary convection-diffusion equation with nonlinear convective as well as diffusive terms which represents a model problem for the solution of the system of the compressible Navier-Stokes equations describing a motion of viscous compressible fluids. We present a discretization of this model equation by the discontinuous Galerkin finite element method. Moreover, under some assumptions on the nonlinear terms, domain partitions and the regularity of the exact solution, we introduce a priori error estimates in the \(L^\infty(0,T;L^2(\Omega))\)-norm and in the \(L^2(0,T;H^1(\Omega))\)-seminorm. A sketch of the proof is presented.
For the entire collection see [Zbl 1194.65013].

MSC:

35Q35	PDEs in connection with fluid mechanics
65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76M10	Finite element methods applied to problems in fluid mechanics
76N15	Gas dynamics (general theory)