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Article

MSC: 65M60, 65M99
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Keywords:
nonstationary nonlinear convection-diffusion equations; time-dependent domain; ALE method; space-time discontinuous Galerkin method; unconditional stability
Summary:
The paper is concerned with the analysis of the space-time discontinuous Galerkin method (STDGM) applied to the numerical solution of the nonstationary nonlinear convection-diffusion initial-boundary value problem in a time-dependent domain formulated with the aid of the arbitrary Lagrangian-Eulerian (ALE) 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 nonlinear convection terms are discretized with the aid of a numerical flux. The space discretization uses piecewise polynomial approximations of degree not greater than $p$ with an integer $p\geq 1$. In the theoretical analysis, the piecewise linear time discretization is used. The main attention is paid to the investigation of unconditional stability of the method.
References:
[1] Akrivis, G., Makridakis, C.: Galerkin time-stepping methods for nonlinear parabolic equations. M2AN, Math. Model. Numer. Anal. 38 (2004), 261-289. DOI 10.1051/m2an:2004013 | MR 2069147 | Zbl 1085.65094
[2] Arnold, D. N., Brezzi, F., Cockburn, B., Marini, L. D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002), 1749-1779. DOI 10.1137/S0036142901384162 | MR 1885715 | Zbl 1008.65080
[3] Babuška, I., Baumann, C. E., Oden, J. T.: A discontinuous {$hp$} finite element method for diffusion problems: 1-D analysis. Comput. Math. Appl. 37 (1999), 103-122. DOI 10.1016/S0898-1221(99)00117-0 | MR 1688050 | Zbl 0940.65076
[4] Balázsová, M., Feistauer, M., Hadrava, M., Kosík, A.: On the stability of the space-time discontinuous Galerkin method for the numerical solution of nonstationary nonlinear convection-diffusion problems. (to appear) in J. Numer. Math.
[5] Bassi, F., Rebay, S.: 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 10.1006/jcph.1996.5572 | MR 1433934 | Zbl 0871.76040
[6] Baumann, C. E., Oden, J. T.: A discontinuous {$hp$} finite element method for the Euler and Navier-Stokes equations. Int. J. Numer. Methods Fluids 31 (1999), 79-95. DOI 10.1002/(SICI)1097-0363(19990915)31:1<79::AID-FLD956>3.0.CO;2-C | MR 1714511 | Zbl 0985.76048
[7] Boffi, D., Gastaldi, L., Heltai, L.: Numerical stability of the finite element immersed boundary method. Math. Models Methods Appl. Sci. 17 (2007), 1479-1505. DOI 10.1142/S0218202507002352 | MR 2359913 | Zbl 1186.76661
[8] Bonito, A., Kyza, I., Nochetto, R. H.: Time-discrete higher-order ALE formulations: stability. SIAM J. Numer. Anal. 51 (2013), 577-604. DOI 10.1137/120862715 | MR 3033024 | Zbl 1267.65114
[9] Brezzi, F., Manzini, G., Marini, D., Pietra, P., Russo, A.: Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differ. Equations 16 (2000), 365-378. DOI 10.1002/1098-2426(200007)16:4<365::AID-NUM2>3.0.CO;2-Y | MR 1765651 | Zbl 0957.65099
[10] Česenek, J., Feistauer, M.: 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 10.1137/110828903 | MR 2970739 | Zbl 1312.65157
[11] Česenek, J., Feistauer, M., Horáček, J., Kučera, V., Prokopová, J.: Simulation of compressible viscous flow in time-dependent domains. Appl. Math. Comput. 219 (2013), 7139-7150. DOI 10.1016/j.amc.2011.08.077 | MR 3030556
[12] Česenek, J., Feistauer, M., Kosík, A.: DGFEM for the analysis of airfoil vibrations induced by compressible flow. ZAMM, Z. Angew. Math. Mech. 93 (2013), 387-402. DOI 10.1002/zamm.201100184 | MR 3069914 | Zbl 1277.74026
[13] Chrysafinos, K., Walkington, N. J.: Error estimates for the discontinuous Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 44 (2006), 349-366. DOI 10.1137/030602289 | MR 2217386 | Zbl 1112.65086
[14] Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16 (2001), 173-261. DOI 10.1023/A:1012873910884 | MR 1873283 | Zbl 1065.76135
[15] Dolejší, V.: On the discontinuous Galerkin method for the numerical solution of the Navier-Stokes equations. Int. J. Numer. Methods Fluids 45 (2004), 1083-1106. DOI 10.1002/fld.730 | MR 2072224 | Zbl 1060.76570
[16] Dolejší, V., Feistauer, M.: Discontinuous Galerkin Method---Analysis and Applications to Compressible Flow. Springer, Heidelberg (2015).
[17] Dolejší, V., Feistauer, M., Hozman, J.: Analysis of semi-implicit DGFEM for nonlinear convection-diffusion problems on nonconforming meshes. Comput. Methods Appl. Mech. Eng. 196 (2007), 2813-2827. DOI 10.1016/j.cma.2006.09.025 | MR 2325393 | Zbl 1121.76033
[18] Donea, J., Giuliani, S., Halleux, J. P.: An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions. Comput. Methods Appl. Mech. Eng. 33 (1982), 689-723. DOI 10.1016/0045-7825(82)90128-1 | Zbl 0508.73063
[19] Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Computational differential equations. Cambridge Univ. Press, Cambridge (1996). MR 1414897 | Zbl 0946.65049
[20] Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems. I. A linear model problem. SIAM J. Numer. Anal. 28 (1991), 43-77. DOI 10.1137/0728003 | MR 1083324 | Zbl 0732.65093
[21] Estep, D., Larsson, S.: The discontinuous Galerkin method for semilinear parabolic problems. RAIRO, Modélisation Math. Anal. Numér. 27 (1993), 35-54. MR 1204627 | Zbl 0768.65065
[22] Feistauer, M., Felcman, J., Straškraba, I.: Mathematical and Computational Methods for Compressible Flow. Numerical Mathematics and Scientific Computation Oxford University Press, Oxford (2003). MR 2261900 | Zbl 1028.76001
[23] Feistauer, M., Hájek, J., Švadlenka, K.: Space-time discontinuous Galerkin method for solving nonstationary convection-diffusion-reaction problems. Appl. Math., Praha 52 (2007), 197-233. DOI 10.1007/s10492-007-0011-8 | MR 2316153 | Zbl 1164.65469
[24] Feistauer, M., Hasnedlová-Prokopová, J., Horáček, J., Kosík, A., Kučera, V.: DGFEM for dynamical systems describing interaction of compressible fluid and structures. J. Comput. Appl. Math. 254 (2013), 17-30. DOI 10.1016/j.cam.2013.03.028 | MR 3061063 | Zbl 1290.65089
[25] Feistauer, M., Horáček, J., Kučera, V., Prokopová, J.: On numerical solution of compressible flow in time-dependent domains. Math. Bohem. 137 (2012), 1-16. MR 2978442 | Zbl 1249.65196
[26] Feistauer, M., Kučera, V., Najzar, K., Prokopová, J.: Analysis of space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Math. 117 (2011), 251-288. DOI 10.1007/s00211-010-0348-x | MR 2754851 | Zbl 1211.65125
[27] Feistauer, M., Kučera, V., Prokopová, J.: Discontinuous Galerkin solution of compressible flow in time-dependent domains. Math. Comput. Simul. 80 (2010), 1612-1623. DOI 10.1016/j.matcom.2009.01.020 | MR 2647255
[28] Formaggia, L., Nobile, F.: A stability analysis for the arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math. 7 (1999), 105-131. MR 1699243 | Zbl 0942.65113
[29] Gastaldi, L.: A priori error estimates for the arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math. 9 (2001), 123-156. MR 1836870 | Zbl 0988.65082
[30] Hasnedlová, J., Feistauer, M., Horáček, J., Kosík, A., Kučera, V.: Numerical simulation of fluid-structure interaction of compressible flow and elastic structure. Computing 95 (2013), S343--S361. DOI 10.1007/s00607-012-0240-x | MR 3054377
[31] Havle, O., Dolejší, V., Feistauer, M.: Discontinuous Galerkin method for nonlinear convection-diffusion problems with mixed Dirichlet-Neumann boundary conditions. Appl. Math., Praha 55 (2010), 353-372. DOI 10.1007/s10492-010-0012-x | MR 2737717 | Zbl 1224.65219
[32] Houston, P., Schwab, C., Süli, E.: Discontinuous $hp$-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002), 2133-2163. DOI 10.1137/S0036142900374111 | MR 1897953 | Zbl 1015.65067
[33] Khadra, K., Angot, P., Parneix, S., Caltagirone, J.-P.: Fictiuous domain approach for numerical modelling of Navier-Stokes equations. Int. J. Numer. Methods Fluids 34 (2000), 651-684. DOI 10.1002/1097-0363(20001230)34:8<651::AID-FLD61>3.0.CO;2-D | Zbl 1032.76041
[34] Oden, J. T., Babuška, I., Baumann, C. E.: A discontinuous $hp$ finite element method for diffusion problems. J. Comput. Phys. 146 (1998), 491-519. DOI 10.1006/jcph.1998.6032 | MR 1654911 | Zbl 0926.65109
[35] Schötzau, D.: hp-DGFEM for Parabolic Evolution Problems. Applications to Diffusion and Viscous Incompressible Fluid Flow. PhD Thesis, ETH No. 13041, Zürich (1999). MR 2715264
[36] Schötzau, D., Schwab, C.: An $hp$ a priori error analysis of the DG time-stepping method for initial value problems. Calcolo 37 (2000), 207-232. DOI 10.1007/s100920070002 | MR 1812787
[37] Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics 25 Springer, Berlin (2006). MR 2249024 | Zbl 1105.65102
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