A priori error estimates of an extrapolated space-time discontinuous galerkin method for nonlinear convection-diffusion problems

Abstract

We deal with the numerical solution of a scalar nonstationary nonlinear convection-diffusion equation. We employ a combination of the discontinuous Galerkin finite element (DGFE) method for the space as well as time discretization. The linear diffusive and penalty terms are treated implicitly whereas the nonlinear convective term is treated by a special higher order explicit extrapolation from the previous time step, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the L ∞ (L 2 )-norm and the L 2 (H 1 )-seminorm with respect to the mesh size h and time step τ . Finally, we present an efficient solution strategy and numerical examples verifying the theoretical results.

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.…”

Optimal spatial error estimates for DG time discretizations

“…The advantage of the semi-implicit approach is that we do not need to solve the nonlinear problem at each time level. Similar technique with nonsymmetric space discretization (NIPG) is described in [25].…”

“…It is easy to see that we are able to extend this assumption from a constant boundary condition with respect to time to a boundary condition which is piecewise polynomial up to the degree q with respect to time. These extensions are described in [25,Remark 4.4]. For details see also [11].…”

Residual based error estimates for the space–time discontinuous Galerkin method applied to the compressible flows

“…On the other hand, the STDG method is very expensive, since the resulting algebraic systems are several times larger (depending on the polynomial approximation degree with respect to time) in comparison to the BDF methods. However, its computational costs are partly compensated by its higher accuracy with respect to time, allowing to use larger time steps, see [14], where numerical experiments indicate that the STDG method requires about 50% of additional computational time in comparison to the BDF scheme, whereas the corresponding algebraic system is four times larger. The STDG method for the compressible flow problems was developed in the series of papers by van der Vegt, van der Ven et al ( [15,16,17,18,19]…”