# On the uniform stability of the space-time discontinuous Galerkin method for nonstationary problems in time-dependent domains

## Main Article Content

## Abstract

In this paper we investigate the stability of the space-time discontinuous Galerkin method (STDGM) for the solution of nonstationary, linear convection-diffusion-reaction problem in time-dependent domains formulated with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. The stability is uniform with respect to the diffusion coefficient. The ALE method replaces the classical partial time derivative with the so called ALE-derivative and an additional convective term. In the second part of the paper we discretize our problem using the space-time discontinuous Galerkin method. In the formulation of the numerical scheme we use the nonsymmetric, symmetric and incomplete versions of the space discretization of diffusion terms and interior and boundary penalty. The space discretization uses piecewise polynomial approximations of degree $p\geq 1$, in time we use only piecewise linear discretization. Finally in the third part of the paper we present our results concerning the uniform unconditional stability of the method.

## Article Details

How to Cite

BALÁZSOVÁ, Monika; FEISTAUER, Miloslav.
On the uniform stability of the space-time discontinuous Galerkin method for nonstationary problems in time-dependent domains.

**Proceedings of the Conference Algoritmy**, [S.l.], p. 84-92, feb. 2016. Available at: <http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/397>. Date accessed: 22 sep. 2017.
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## References

[1] M. Balázsova, M. Feistauer, On the stability of the ALE space-time discontinuous Galerkin method for the numerical solution of nonlinear convection-diffusion problems in timedependent domains, Appl. Math. 60 (2015) No.5, pp. 501–526

[2] M. Balázsova, M. Feistauer, M. Hadrava and A. Kosík, On the stability of the spacetime discontinuous Galerkin method for the numerical solution of nonstationary nonlinear convection-diffusion problems, J. Numer. Math., 23 (3) (2015), pp. 211–233.

[3] A. Bonito, I. Kyza, R.H. Nochetto, Time-discrete higher-order ALE formulations: Stability, SIAM J. Numer. Anal. 51 (1) (2013), pp. 577–604.

[4] V. Dolejší, M. Feistauer, Discontinuous Galerkin method – Analysis and applications to Compressible Flow, Springer, Heidelberg, (2015).

[5] M. Feistauer, J. Felcman, I. Straškraba, Mathematical and Computational Methods for Compressible Flow, Clarendon Press, Oxford, 2003.

[6] M. Feistauer, J. Hájek, K. Švadlenka, Space-time discontinuous Galerkin method for solving nonstationary linear convection-diffusion-reaction problems, Appl. Math. 52 (2007), pp. 197–233.

[7] T. Nomura and T.J.R. Hughes, An arbitrary Lagrangian-Eulerian finite element method for interaction of fluid and a rigid body, Comput. Methods Appl. Mech. Engrg., 95 (1992), pp. 115-138.

[2] M. Balázsova, M. Feistauer, M. Hadrava and A. Kosík, On the stability of the spacetime discontinuous Galerkin method for the numerical solution of nonstationary nonlinear convection-diffusion problems, J. Numer. Math., 23 (3) (2015), pp. 211–233.

[3] A. Bonito, I. Kyza, R.H. Nochetto, Time-discrete higher-order ALE formulations: Stability, SIAM J. Numer. Anal. 51 (1) (2013), pp. 577–604.

[4] V. Dolejší, M. Feistauer, Discontinuous Galerkin method – Analysis and applications to Compressible Flow, Springer, Heidelberg, (2015).

[5] M. Feistauer, J. Felcman, I. Straškraba, Mathematical and Computational Methods for Compressible Flow, Clarendon Press, Oxford, 2003.

[6] M. Feistauer, J. Hájek, K. Švadlenka, Space-time discontinuous Galerkin method for solving nonstationary linear convection-diffusion-reaction problems, Appl. Math. 52 (2007), pp. 197–233.

[7] T. Nomura and T.J.R. Hughes, An arbitrary Lagrangian-Eulerian finite element method for interaction of fluid and a rigid body, Comput. Methods Appl. Mech. Engrg., 95 (1992), pp. 115-138.