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A priori error estimates of an extrapolated space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. (English) Zbl 1237.65105
The numerical solution of a scalar nonstationary nonlinear convection-diffusion equation for 2D or 3D problems is studied. The aim is to develop a sufficiently efficient, robust and accurate numerical scheme for the simulation of unsteady viscous compressible flows. For the numerical scheme a combination of the discontinuous Galerkin finite element method for the space as well for the time discretization is used. In the numerical scheme, the linear diffusive and penalty terms are computed implicitly and the nonlinear convective term is treated in the previous time step. For this purpose a special higher order explicit extrapolation is used. A priori asymptotic error estimates in \( L_\infty((0,T), L_2(\Omega)) \) and \(L_2((0,T), H^1(\Omega))\) functional spaces under some relation on the mesh size \(h\) and the time step \( \tau\) are proved. Several numerical experiments together with the implementation notes for the proposed scheme conclude the paper. These numerical experiments verify the theoretical results.

MSC:
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
76N15 Gas dynamics (general theory)
76M10 Finite element methods applied to problems in fluid mechanics
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