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Space-time discontinuous Galerkin method for solving nonstationary convection-diffusion-reaction problems. (English) Zbl 1164.65469
Summary: The paper presents the theory of the discontinuous Galerkin finite element method for the space-time discretization of a linear nonstationary convection-diffusion-reaction initial-boundary value problem. The discontinuous Galerkin method is applied separately in space and time using, in general, different nonconforming space grids on different time levels and different polynomial degrees \(p\) and \(q\) in space and time discretization, respectively. In the space discretization, the nonsymmetric interior and boundary penalty approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “\(L^2(L^2)\)”- and “\( \sqrt { \varepsilon } L^2(H^1) \)”-norms, where \(\varepsilon \geq 0\) is the diffusion coefficient. Using special interpolation theorems for the space as well as time discretization, we find that under some assumptions on the shape regularity of the meshes and a certain regularity of the exact solution the errors are of order \( O(h^p+\tau ^q)\). The estimates hold true even in the hyperbolic case when \( \varepsilon = 0\).

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
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
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