On the Time Growth of the Error of the Discontinuous Galerkin Method for Advection-reaction Problems
Abstrakt
This contribution presents an overview of the results of the paper Kučera & Shu (2019) on the time growth of the error of the discontinuous Galerkin (DG) method. When estimating quantities of interest in differential equations, the application of Gronwall's lemma gives estimates which grow exponentially in time even for problems where such behavior is unnatural.
In the case of a non-stationary advection-diffusion equation we can circumvent this problem by considering a general space-time exponential scaling argument. Thus we obtain error estimates for DG which grow exponentially not in time, but in the time particles carried by the flow field spend in the spatial domain.
If this is uniformly bounded, one obtains an error estimate of the form $C(h^{p+1/2})$, where p is the degree of polynomials used in the DG method and C is independent of time. We discuss the time growth of the exact solution and the exponential scaling argument and give an overview of results from Kučera & Shu (2019) and the tools necessary for the analysis.
