Stability of the ale space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 6, pp. 2327-2356.

The paper is concerned with the analysis of the space-time discontinuous Galerkin method (STDGM) applied to the numerical solution of nonstationary nonlinear convection-diffusion initial- boundary value problem in a time-dependent domain. The problem is reformulated 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. The problem is discretized with the use of the ALE- space time discontinuous Galerkin method (ALE-STDGM). 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 nonlinear convection terms are discretized with the aid of a numerical flux. The main attention is paid to the proof of 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.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2018062
Classification : 65M60, 65M99
Mots clés : nonlinear convection-diffusion equation, time-dependent domain, ALE method, space-time discontinuous Galerkin method, discrete characteristic function, unconditional stability in space and time
Balázsová, Monika 1 ; Feistauer, Miloslav 1 ; Vlasák, Miloslav 1

1
@article{M2AN_2018__52_6_2327_0,
     author = {Bal\'azsov\'a, Monika and Feistauer, Miloslav and Vlas\'ak, Miloslav},
     title = {Stability of the ale space-time discontinuous {Galerkin} method for nonlinear convection-diffusion problems in time-dependent domains},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {2327--2356},
     publisher = {EDP-Sciences},
     volume = {52},
     number = {6},
     year = {2018},
     doi = {10.1051/m2an/2018062},
     zbl = {1417.65166},
     mrnumber = {3905191},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2018062/}
}
TY  - JOUR
AU  - Balázsová, Monika
AU  - Feistauer, Miloslav
AU  - Vlasák, Miloslav
TI  - Stability of the ale space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2018
SP  - 2327
EP  - 2356
VL  - 52
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2018062/
DO  - 10.1051/m2an/2018062
LA  - en
ID  - M2AN_2018__52_6_2327_0
ER  - 
%0 Journal Article
%A Balázsová, Monika
%A Feistauer, Miloslav
%A Vlasák, Miloslav
%T Stability of the ale space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2018
%P 2327-2356
%V 52
%N 6
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2018062/
%R 10.1051/m2an/2018062
%G en
%F M2AN_2018__52_6_2327_0
Balázsová, Monika; Feistauer, Miloslav; Vlasák, Miloslav. Stability of the ale space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 6, pp. 2327-2356. doi : 10.1051/m2an/2018062. http://www.numdam.org/articles/10.1051/m2an/2018062/

[1] G. Akrivis and C. Makridakis, Galerkin time-stepping methods for nonlinear parabolic equations. ESAIM: M2AN 38 (2004) 261–289. | DOI | Numdam | MR | Zbl

[2] D.N. Arnold, F. Brezzi, B. Cockburn and D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749–1779. | DOI | MR | Zbl

[3] I. Babuška, C.E. Baumann and T.J. Oden, A discontinuous hp finite element method for diffusion problems, 1D analysis. Comput. Math. Appl. 37 (1999) 103–122. | DOI | MR | Zbl

[4] S. Badia and R. Codina, Analysis of a stabilized finite element approximation of the transient convection-diffusion equation using an ALE framework. SIAM J. Numer. Anal. 44 (2006) 2159–2197. | DOI | MR | Zbl

[5] M. Balázsová and M. Feistauer, On the stability of the space-time discontinuous Galerkin method for nonlinear convection- diffusion problems in time-dependent domains. Appl. Math. 60 (2015) 501–526. | DOI | MR | Zbl

[6] M. Balázsová and M. Feistauer, On the uniform stability of the space-time discontinuous Galerkin method for nonstationary problems in time-dependent domains. Proc. Conf. ALGORITMY (2016) 84–92.

[7] M. Balázsová, M. Feistauer, M. Hadrava and A. Kosík, On the stability of the space-time discontinuous Galerkin method for the numerical solution of nonstationary nonlinear convection-diffusion problems. J. Numer. Math. 23 (2015) 211–233. | DOI | MR | Zbl

[8] F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131 (1997) 267–279. | DOI | MR | Zbl

[9] C.E. Baumann and T.J. Oden, A discontinuous hp finite element method for the Euler and Navier-Stokes equations. Int. J. Numer. Methods Fluids 31 (1999) 79–95. | DOI | MR | Zbl

[10] D. Boffi, L. Gastaldi and L. Heltai, Numerical stability of the finite element immersed boundary method. Math. Models Methods Appl. Sci. 17 (2007) 1479–1505. | DOI | MR | Zbl

[11] A. Bonito, I. Kyza, R.H. Nochetto, Time-discrete higher-order ALE formulations: stability. SIAM J. Numer. Anal. 51 (2013) 577–604. | DOI | MR | Zbl

[12] A. Bonito, I. Kyza and R.H. Nochetto, Time-discrete higher order ALE formulations: a priori error analysis. Numer. Math. 125 (2013) 225–257. | DOI | MR | Zbl

[13] F. Brezzi, G. Manzini, D. Marini, P. Pietra and A. Russo, Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differ. Equ. 16 (2000) 365–378. | DOI | MR | Zbl

[14] J. Česenek and M. Feistauer, Theory of the space-time discontinuous Galerkin method for nonstationary parabolic problems with nonlinear convection and diffusion. SIAM J. Numer. Anal. 50 (2012) 1181–1206. | DOI | MR | Zbl

[15] J. Česenek, M. Feistauer, J. Horáček, V. Kučera and J. Prokopová, Simulation of compressible viscous flow in time-dependent domains. Appl. Math. Comput. 219 (2013) 7139–7150. | DOI | MR | Zbl

[16] J. Česenek, M. Feistauer and A. Kosík, DGFEM for the analysis of airfoil vibrations induced by compressible flow. ZAMM Z. Angew. Math. Mech. 93 (2013) 387–402. | DOI | MR | Zbl

[17] K. Chrysafinos and N.J. Walkington, Error estimates for the discontinuous Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 44 (2006) 349–366. | DOI | MR | Zbl

[18] A. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. Review article. J. Sci. Comput. 16 (2001) 173–261. | MR | Zbl

[19] V. Dolejší, On the discontinuous Galerkin method for the numerical solution of the Navier-Stokes equations. Int. J. Numer. Methods Fluids 45 (2004) 1083–1106. | DOI | MR | Zbl

[20] V. Dolejší and M. Feistauer. Discontinuous Galerkin Method – Analysis and Applications to Compressible Flow. Springer, Berlin (2015). | DOI | MR

[21] V. Dolejší, M. Feistauer and J. Hozman, Analysis of semi-implicit DGFEM for nonlinear convection-diffusion problems on nonconforming meshes. Comput. Methods Appl. Mech. Eng. 196 (2007) 2813–2827. | DOI | MR | Zbl

[22] J. Donéa, S. Giuliani and J. Halleux, An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid- structure interactions. Comput. Methods Appl. Mech. Eng. 33 (1982) 689–723. | DOI | Zbl

[23] K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Computational Differential Equations. Cambridge University Press, Cambridge (1996). | MR | Zbl

[24] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems I: a linear model problem. SIAM J. Numer. Anal. 28 (1991) 43–77. | DOI | MR | Zbl

[25] D. Estep and S. Larsson, The discontinuous Galerkin method for semilinear parabolic problems. ESAIM: M2AN 27 (1993) 35–54. | DOI | Numdam | MR | Zbl

[26] M. Feistauer, V. Dolejší and V. Kučera, On the discontinuous Galerkin method for the simulation of compressible flow with wide range of Mach numbers. Comput. Visual. Sci. 10 (2007) 17–27. | DOI | MR

[27] M. Feistauer, J. Felcman and I. Straškraba, Mathematical and Computational Methods for Compressible Flow. Clarendon Press, Oxford (2003). | MR | Zbl

[28] M. Feistauer, M. Hadrava, J. Horáček and A. Kosík, Numerical solution of fluid-structure interaction by the space-time discontinuous Galerkin method, edited by J. Fuhrmann, M. Ohlberger and C. Rohde. In: Proc. of the conf. FVCA7 (Finite volumes for complex applications VII elliptic, parabolic and hyperbolic problems), Berlin, June 16–20. Springer, Cham (2014) 567–575. | DOI | MR

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

[30] M. Feistauer, J. Hasnedlová-Prokopová, J. Horáček, A. Kosík and V. Kučera, DGFEM for dynamical systems describing interaction of compressible fluid and structures. J. Comput. Appl. Math. 254 (2013) 17–30. | DOI | MR | Zbl

[31] M. Feistauer, J. Horáček, V. Kučera and J. Prokopová, On the numerical solution of compressible flow in time-dependent domains. Math. Bohem. 137 (2012) 1–16. | DOI | MR | Zbl

[32] M. Feistauer and V. Kuscera, On a robust discontinuous Galerkin technique for the solution of compressible flow. J. Comput. Phys. 224 (2007) 208–221. | DOI | MR | Zbl

[33] M. Feistauer, V. Kučera, K. Najzar and J. Prokopová, Analysis of space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Math. 117 (2011) 251–288. | DOI | MR | Zbl

[34] M. Feistauer, V. Kučera and J. Prokopová, Discontinuous Galerkin solution of compressible flow in time-dependent domains. Math. Comput. Simul. 80 (2010) 1612–1623. | DOI | MR | Zbl

[35] L. Formaggia and F. Nobile, A stability analysis for the arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math. 7 (1999) 105–131. | MR | Zbl

[36] L. Gastaldi, A priori error estimates for the Arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math. 9 (2001) 123–156. | MR | Zbl

[37] J. Hasnedlová, M. Feistauer, J. Horáček, A. Kosík and V. Kučera, Numerical simulation of fluid-structure interaction of compressible flow and elastic structure. Computing 95 (2013) 343–361. | DOI | MR

[38] O. Havle, V. Dolejší and M. Feistauer, Discontinuous Galerkin method for nonlinear convection-diffusion problems with mixed Dirichlet-Neumann boundary conditions. Appl. Math. 55 (2010) 353–372. | DOI | MR | Zbl

[39] C.W. Hirt, A.A. Amsdem and J.L. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys. 135 (1997) 198–216. | MR | Zbl

[40] P. Houston, C. Schwab and E. Süli, Discontinuous hp-finite element methods for advection-diffusion problems. SIAM J. Numer. Anal. 39 (2002) 2133–2163. | DOI | MR | Zbl

[41] T.J.R. Hughes, W.K. Liu and T.K. Zimmermann, Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput. Methods Appl. Mech. Eng. 29 (1981) 329–349. | DOI | MR | Zbl

[42] 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. Eng. 95 (1992) 115–138. | DOI | Zbl

[43] K. Khadra, P. Angot, S. Parneix and J.-P. Caltagirone, Fictitious domain approach for numerical modelling of Navier-Stokes equations. Int. J. Numer. Methods Fluids 34 (2000) 651–684. | DOI | Zbl

[44] A. Kosík, M. Feistauer, M. Hadrava and J. Horáček, Numerical simulation of the interaction between a nonlinear elastic structure and compressible flow by the discontinuous Galerkin method. Appl. Math. Comput. 267 (2015) 382–396. | DOI | MR | Zbl

[45] V. Kučera and M. Vlasák, A priori diffusion-uniform error estimates for nonlinear singularly perturbed problems: BDF2, midpoint and time DG. ESAIM: M2AN 51 (2017) 537–563. | DOI | Numdam | MR | Zbl

[46] J.T. Oden, I. Babuska and C.E. Baumann, A discontinuous hp finite element method for diffusion problems. J. Comput. Phys. 146 (1998) 491–519. | DOI | MR | Zbl

[47] C.-W. Shu, Discontinuous Galerkin method for time dependent problems: survey and recent developments, edited by X. Feng, et al. Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations. Springer, Cham (2014) 25–62. | DOI | MR | Zbl

[48] D. Schötzau, hp-DGFEM for parabolic evolution problems. Applications to diffusion and viscous incompressible fluid flow. Ph.D. thesis, ETH No. 13041, Zürich (1999). | MR

[49] D. Schiötzau and C. Schwab, An hp a priori error analysis of the Discontinuous Galerkin time-stepping method for initial value problems. Calcolo 37 (2000) 207–232. | DOI | MR | Zbl

[50] V. Thomaée, Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006). | Zbl

[51] J.J.W. Van Der Vegt and H. Van Der Ven, Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows. Part I. General formulation. J. Comput. Phys. 182 (2002) 546–585. | MR | Zbl

[52] M. Vlasák, Optimal spatial error estimates for DG time discretizations. J. Numer. Math. 21 (2013) 201–230. | DOI | MR | Zbl

[53] M. Vlasák, V. Dolejší and J. Hájek, A Priori error estimates of an extrapolated space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Methods Partial Differ. Equ. 27 (2011) 1456–1482. | DOI | MR | Zbl

[54] 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 (2004) 641–666. | DOI | MR | Zbl

Cité par Sources :