Analysis of space–time discontinuous Galerkin method for nonlinear convection–diffusion problems

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.

L2-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

“…References [5,24,25,28] present results analogous to those in [2,18], but for a combined FE-FV method involving piecewise linear conforming finite elements and dual finite volumes (triangular finite volumes in the case of [5]). Similar L 2 (H 1 )-and L ∞ (L 2 )-error estimates as in [18] are shown in [27,50], but with respect to various discontinuous Galerkin schemes.…”

A space-time discontinuous Galerkin method for Boussinesq-type equations

“…The aim of this article is therefore to construct a new numerical method for Boussinesq-type equations that is very high order accurate in both space and time, but without such a cubic time-step restriction in the presence of dispersive terms. For this purpose we propose an implicit method that is based on the space-time DG method, which was first introduced by Van der Vegt et al in [28,36,37] and which has been further analyzed for convection-diffusion problems in the work of Feistauer et al [9,23].…”

Anisotropic hp-adaptive discontinuous Galerkin method for the numerical solution of time dependent PDEs

“…The density q is used as the quantity u in (33) and in condition (P1 H ) of Problem 5.1 We carried out the computations with quadratic approximation with respect to the time (k ¼ 2 in (30)) and with the tolerances x ¼ 8 Á 10 À4 ; x ¼ 4 Á 10 À4 , x ¼ 2 Á 10 À4 and x ¼ 10 À4 .…”

“…The function ðu h Þ À 0 is given by the initial condition. For more details see, e.g., [32,33]. The coupling between two time levels is performed by the second term in (31) …”