# On the time growth of the error of the discontinuous Galerkin method for advection-reaction problems

## Main Article Content

## Abstract

This contribution presents an overview of the results of the paper [6] 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(hp+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 [6] and the tools necessary for the analysis.

## Article Details

How to Cite

Kučera, V.
(2020).
On the time growth of the error of the discontinuous Galerkin method for advection-reaction problems.

*Proceedings Of The Conference Algoritmy,*, 221 - 228. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/1578/830
Section

Articles

## References

[1] B. Ayuso and L. D. Marini, Discontinuous Galerkin methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal. 47:2 (2009), 1391–1420.

[2] A. Devinatz, R. Ellis, and A. Friedman, The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives, II, Indiana Univ. Math. J. 23 (1974), 991–1011.

[3] M. Feistauer and K. Švadlenka, Discontinuous Galerkin method of lines for solving non- stationary singularly perturbed linear problems, J. Numer. Math. 12:2 (2004), 97–117.

[4] C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp. 46:173 (1986), 1–26.

[5] V. Kučera, On diffusion-uniform error estimates for the DG method applied to singularly perturbed problems, IMA J. Numer. Anal. 34:2 (2014), 820–861.

[6] V. Kučera and C.-W. Shu, On the time growth of the error of the DG method for advective problems, IMA J. Numer. Anal. 39:2 (2019) 687–712.

[7] U. Nävert, A finite element method for convection-diffusion problems, Ph.D. thesis, Chalmers University of Technology, 1982.

[8] W. H. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Report LA-UR-73-479 (1973).

[9] Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J. Numer. Anal. 42:2 (2004), 641–666.

[2] A. Devinatz, R. Ellis, and A. Friedman, The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives, II, Indiana Univ. Math. J. 23 (1974), 991–1011.

[3] M. Feistauer and K. Švadlenka, Discontinuous Galerkin method of lines for solving non- stationary singularly perturbed linear problems, J. Numer. Math. 12:2 (2004), 97–117.

[4] C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp. 46:173 (1986), 1–26.

[5] V. Kučera, On diffusion-uniform error estimates for the DG method applied to singularly perturbed problems, IMA J. Numer. Anal. 34:2 (2014), 820–861.

[6] V. Kučera and C.-W. Shu, On the time growth of the error of the DG method for advective problems, IMA J. Numer. Anal. 39:2 (2019) 687–712.

[7] U. Nävert, A finite element method for convection-diffusion problems, Ph.D. thesis, Chalmers University of Technology, 1982.

[8] W. H. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Report LA-UR-73-479 (1973).

[9] Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J. Numer. Anal. 42:2 (2004), 641–666.