Abstract
The paper presents the theory of the discontinuous Galerkin finite element method for the space–time discretization of a nonstationary convection–diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The problem is not singularly perturbed with dominating convection. The discontinuous Galerkin method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time dicretization. In the space discretization the nonsymmetric, symmetric and incomplete interior and boundary penalty (NIPG, SIPG, IIPG) approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L 2(L 2)”- and “DG”-norm formed by the “L 2(H 1)”-seminorm and penalty terms. A special technique based on the use of the Gauss–Radau interpolation and numerical integration has been used for the derivation of an abstract error estimate. In the “DG”-norm the error estimates are optimal with respect to the size of the space grid. They are optimal with respect to the time step, if the Dirichlet boundary condition has behaviour in time as a polynomial of degree ≤ q.
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This work was a part of the project No. MSM 0021620839 financed by the Ministry of Education of the Czech Republic and was partly supported under the Grant No. 201/08/0012 of the Czech Grant Agency of the Czech Republic. The research of J. Prokopová was supported under the Grant No. 12810 of the Grant Agency of the Charles University in Prague.
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Feistauer, M., Kučera, V., Najzar, K. et al. Analysis of space–time discontinuous Galerkin method for nonlinear convection–diffusion problems. Numer. Math. 117, 251–288 (2011). https://doi.org/10.1007/s00211-010-0348-x
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DOI: https://doi.org/10.1007/s00211-010-0348-x