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Analysis of space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. (English) Zbl 1211.65125
The authors study the space-time discontinuous Galerkin discretization applied separately in space and in time for the numerical solution of a nonstationary nonlinear convection-diffusion equation [cf. M. Feistauer, J. Hájek and K. Švadlenka, Appl. Math., Praha 52, No. 3, 197–233 (2007; Zbl 1164.65469)]. The main object of the paper is to derive error estimates of the space-time discontinuous Galerkin finite element method for the nonstationary initial-boundary value problem with nonlinear convection and linear diffusion. These estimates are optimal in time if the Dirichlet boundary conditions have behaviour in time as a polynomial of degree $$\leq q$$. In general case these estimates become suboptimal.

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations