Title:
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Stability of ALE space-time discontinuous Galerkin method (English) |
Author:
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Vlasák, Miloslav |
Author:
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Balázsová, Monika |
Author:
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Feistauer, Miloslav |
Language:
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English |
Journal:
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Proceedings of Equadiff 14 |
Volume:
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Conference on Differential Equations and Their Applications, Bratislava, July 24-28, 2017 |
Issue:
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2017 |
Year:
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Pages:
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237-246 |
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Category:
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math |
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Summary:
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We assume the heat equation in a time dependent domain, where the evolution of the domain is described by a given mapping. The problem is discretized by the discontinuous Galerkin (DG) method in space as well as in time with the aid of Arbitrary Lagrangian-Eulerian (ALE) method. The sketch of the proof of the stability of the method is shown. (English) |
Keyword:
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ALE formulation, discontinuous Galerkin method, discrete characteristic function, stability |
MSC:
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65M60 |
MSC:
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65M99 |
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Date available:
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2019-09-27T08:05:47Z |
Last updated:
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2019-09-27 |
Stable URL:
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http://hdl.handle.net/10338.dmlcz/703015 |
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Reference:
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[1] Arnold, D. N., Brezzi, F., Cockburn, B., HASH(0x22e2088), Marini., L. D.: Unified analysis of discontinuous Galerkin methods for elliptic problems.. SIAM J. Numer. Anal., 39(5), 1749–1779, 2002. MR 1885715, 10.1137/S0036142901384162 |
Reference:
|
[2] Balázsová, M., HASH(0x22f8290), Feistauer., M.: 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. MR 3396478, 10.1007/s10492-015-0109-3 |
Reference:
|
[3] Balázsová, M., Feistauer, M., HASH(0x22f8cb0), Vlasák., M.: Stability of the ALE space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains.. (in preparation). MR 3396478 |
Reference:
|
[4] Boffi, D., Gastaldi, L., and, Heltai., L.: Numerical stability of the finite element immersed boundary method.. Math. Models Methods Appl. Sci., 17, 1479–1505, 2007. MR 2359913, 10.1142/S0218202507002352 |
Reference:
|
[5] Bonito, A., Kyza, I., and, Nochetto., R. H.: Time-discrete higher-order ALE formulations: Stability.. SIAM J. Numer. Anal., 51(1), 577–604, 2013. MR 3033024, 10.1137/120862715 |
Reference:
|
[6] Bonito, A., Kyza, I., and, Nochetto., R. H.: Time-discrete higher order ALE formulations: a priori error analysis.. Numer. Math., 125, 225–257, 2013. MR 3101828, 10.1007/s00211-013-0539-3 |
Reference:
|
[7] Česenek, J., Feistauer, M., Horáček, J., Kučera, V., HASH(0x22ffd00), Prokopová., J.: Simulation of compressible viscous flow in time-dependent domains.. Appl. Math. Comput., 219, 7139–7150, 2013. MR 3030556 |
Reference:
|
[8] Česenek, J., Feistauer, M., HASH(0x2301ab8), Kosík., A.: DGFEM for the analysis of airfoil vibrations induced by compressible flow.. Z. Angew. Math. Mech., 93 No. 6-7, 387–402, 2013. MR 3069914, 10.1002/zamm.201100184 |
Reference:
|
[9] Chrysafinos, K., HASH(0x23023b8), Walkington., N. J.: Error estimates for the discontinuous Galerkin methods for parabolic equations.. SIAM J. Numer. Anal., 44, 349–366, 2006. MR 2217386, 10.1137/030602289 |
Reference:
|
[10] Cockburn, B., Karniadakis, G. E., HASH(0x2306760), Shu., C.-W.: Discontinuous Galerkin methods.. In Lecture Notes in Computational Science and Engineering 11. Springer, Berlin, 2000. MR 1842160 |
Reference:
|
[11] Dolejší, V., Feistauer., M.: Discontinuous Galerkin method, Analysis and applications to compressible flow.. Cham: Springer, 2015. MR 3363720 |
Reference:
|
[12] Ehle., B. L.: On Padé approximations to the exponential function and A-stable methods for the numerical solution of initial value problems.. Research report CSRR 2010, Dept. AACS, Univ. of Waterloo, Ontario, Canada, 1969. MR 2716012 |
Reference:
|
[13] Formaggia, L., HASH(0x230ae50), Nobile., F.: A stability analysis for the arbitrary Lagrangian Eulerian formulation with finite elements.. East-West J. Numer. Math., 7(2), 105–131, 1999. MR 1699243 |
Reference:
|
[14] Gastaldi., L.: A priori error estimates for the Arbitrary Lagrangian Eulerian formulation with finite elements.. East-West J. Numer. Math., 9(2), 123–156, 2001. MR 1836870 |
Reference:
|
[15] Guillo, A., HASH(0x230ece0), Soulé., J. L.: La résolution numérique des problèmes différentiels aux conditions initiales par des méthodes de collocation.. R.A.I.R.O., R-3, 17–44, 1969. MR 0280008 |
Reference:
|
[16] Hairer, E., Norsett, S. P., HASH(0x230f700), Wanner., G.: Solving ordinary differential equations I, Nonstiff problems.. Number 8 in Springer Series in Computational Mathematics. Springer Verlag, 2000. MR 1227985 |
Reference:
|
[17] Hairer, E., Wanner., G.: Solving ordinary differential equations II, Stiff and differential algebraic problems.. Springer Verlag, 2002. MR 1439506 |
Reference:
|
[18] Hirt, C. W., Amsdem, A. A., HASH(0x2311730), Cook., J. L.: An arbitrary Lagrangian-Eulerian computing method for all flow speeds.. J. Comput. Phys., 135(2), 198–216, 1997. MR 1486272, 10.1006/jcph.1997.5727 |
Reference:
|
[19] Hughes, T. J. R., Liu, W. K., HASH(0x2314108), Zimmermann., T. K.: Lagrangian-Eulerian finite element formulation for incompressible viscous flows.. Comput. Methods Appl. Mech. Eng., 29(3), 329–349, 1981. MR 0659925, 10.1016/0045-7825(81)90049-9 |
Reference:
|
[20] Hulme., B. L.: One step piecewise polynomial Galerkin methods for initial value problems.. Math. Comp., 26, 415–424, 1972. MR 0321301, 10.1090/S0025-5718-1972-0321301-2 |
Reference:
|
[21] Khadra, K., Angot, P., Parneix, S., HASH(0x23180b8), Caltagirone., J.-P.: Fictitious domain approach for numerical modelling of Navier-Stokes equations.. Int. J. Numer. Methods Fluids, 34(8), 651–684, 2000. 10.1002/1097-0363(20001230)34:8<651::AID-FLD61>3.0.CO;2-D |
Reference:
|
[22] Kosík, A., Feistauer, M., Hadrava, M., HASH(0x231a290), Horáček., J.: 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. MR 3399055 |
Reference:
|
[23] Thomeé., V.: Galerkin finite element methods for parabolic problems.. 2nd revised and expanded. Springer, Berlin, 2006. MR 2249024 |
Reference:
|
[24] Vlasák, M., Dolejší, V., HASH(0x231d550), Hájek., J.: A priori error estimates of an extrapolated space time discontinuous Galerkin method for nonlinear convection-diffusion problems.. Numer. Methods Partial Differ. Equations, 27(6), 1453–1482, 2011. MR 2838303, 10.1002/num.20591 |
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