Abstract
In this paper we investigate the stability of the space-time discontinuous Galerkin method for the solution of nonstationary, nonlinear convection-diffusion problem in time-dependent domains. At first we define the continuous problem and reformulate it using the Arbitrary Lagrangian-Eulerian (ALE) method, which replaces the classical partial time derivative by the so called ALE-derivative and an additional convective term. Then the problem is discretized with the aid of the ALE space-time discontinuous Galerkin method (ALE-STDGM). The discretization uses piecewise polynomial functions of degree p ≥ 1 in space and q > 1 in time. Finally in the last part of the paper we present our results concerning the unconditional stability of the method. An important step is the generalization of a discrete characteristic function associated with the approximate solution and the derivation of its properties, namely its continuity in the \(\Vert \cdot \Vert _{L^2}\)-norm and in special ∥⋅∥DG-norm.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
M. Balázsová, M. Feistauer, On the stability of the space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains. Appl. Math. 60, 501–526 (2015)
M. Balázsová, M. Feistauer, On the uniform stability of the space-time discontinuous Galerkin method for nonstationary problems in time-dependent domains, in ALGORITMY 2016, 20th Conference on Scientific Computing, Vysoké Tatry - Podbanské, Slovakia, March 13–18, 2016, ed. by A. Handlovičová, D. Ševčovič (Slovak University of Technology, Bratislava, 2016), pp. 84–92
J. Česenek, M. Feistauer, J. Horáček, V. Kučera, J. Prokopová, Simulation of compressible viscous flow in time-dependent domains. Appl. Math. Comput. 219, 7139–7150 (2013)
J. Česenek, M. Feistauer, A. Kosík, DGFEM for the analysis of airfoil vibrations induced by compressible flow. ZAMM Z. Angew. Math. Mech. 93(6–7), 387–402 (2013)
V. Dolejší, M. Feistauer, Discontinuous Galerkin method – Analysis and Applications to Compressible Flow (Springer, Berlin, 2015)
A. Kosík, M. Feistauer, M. Hadrava, J. Horáček, Numerical simulation of the interaction between a nonlinear elastic structure and compressible flow by the discontinuous Galerkin method. Appl. Math. Comput. 267, 382–396 (2015)
Acknowledgements
This research was supported by the project GA UK No. 127615 of the Charles University (M. Balázsová) and by the grant 17-01747S of the Czech Science Foundation (M. Vlasák, who is a junior member of the University Centre for Mathematical Modeling, Applied Analysis and Computational Mathematics - MathMAC).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Balázsová, M., Vlasák, M. (2019). Stability of Higher-Order ALE-STDGM for Nonlinear Problems in Time-Dependent Domains. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_51
Download citation
DOI: https://doi.org/10.1007/978-3-319-96415-7_51
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-96414-0
Online ISBN: 978-3-319-96415-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)