×

Residual-based variational multiscale modeling in a discontinuous Galerkin framework. (English) Zbl 1433.65227

This paper develops the general form of the variational multiscale (VMS) finite element formulation in a discontinuous Galerkin (DG) framework. The authors view it as a first step towards a complete integration of VMS in DG. A key feature is that it is irrespective of the continuity of the coarse-scale basis, standing in sharp contrast to the classical approach, where sufficient continuity of the coarse scales is always assumed. A new residual-based VMS formulation in the context of DG is also explored where the authors use the discontinuity on the element interfaces as a measure for the fine-scale boundary values such that the assumption of vanishing fine scales on element boundaries is no longer needed.
The paper demonstrates, for the one-dimensional Poisson problem, that existing discontinuous Galerkin formulations, such as the interior penalty method, can be rederived by choosing particular fine-scale interface models. In addition, it shows, for the one-dimensional advection-diffusion problem, that upwind numerical fluxes can be interpreted as an ad hoc remedy for missing volumetric fine-scale terms.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M80 Fundamental solutions, Green’s function methods, etc. for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] I. Akkerman, Y. Bazilevs, V. M. Calo, T. J. R. Hughes, and S. J. Hulshoff, The role of continuity in residual-based variational multiscale modeling of turbulence, Comput. Mech., 41 (2008), pp. 371–378. · Zbl 1162.76355
[2] D. Arnold, F. Brezzi, B. Cockburn, and D. Marini, Discontinuous Galerkin methods for elliptic problems, in Discontinuous Galerkin Methods. Theory, Computation and Applications, B. Cockburn, G. E. Karniadakis, and C. W. Shu, eds., Lect. Notes Comput. Sci. Eng., Springer-Verlag, Berlin, 11 (2000), pp. 89–101. · Zbl 0948.65127
[3] D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), pp. 1749–1779. · Zbl 1008.65080
[4] I. Babuška and M. Zlámal, Nonconforming elements in the finite element method with penalty, SIAM J. Numer. Anal., 10 (1973), pp. 863–875. · Zbl 0237.65066
[5] 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), pp. 267–279. · Zbl 0871.76040
[6] F. Bassi, S. Rebay, G. Mariotti, S. Pedinotti, and M. Savini, A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows, in Proceedings of the 2nd European Conference on Turbomachinery, Fluid Dynamics and Thermodynamicst, R. Decuypere and G. Dibelius, eds., Antwerpen, Belgium, 1997, pp. 99–108.
[7] C. E. Baumann and J. T. Oden, A discontinuous hp-finite element method for convection-diffusion problems, Comput. Methods. Appl. Mech. Engrg., 175 (1999), pp. 311–341. · Zbl 0924.76051
[8] Y. Bazilevs, Isogeometric Analysis of Turbulence and Fluid-Structure Interaction, Ph.D. thesis, The University of Texas at Austin, Austin, TX, 2006.
[9] Y. Bazilevs, V. M. Calo, J. A. Cottrell, T. J. R. Hughes, A. Reali, and G. Scovazzi, Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Comput. Methods Appl. Mech. Engrg., 197 (2007), pp. 173–201. · Zbl 1169.76352
[10] P. Bochev, T. J. R. Hughes, and G. Scovazzi, A multiscale discontinuous Galerkin method, in Proceedings of the International Conference on Large-Scale Scientific Computing, Springer, 2005, pp. 84–93. · Zbl 1142.65442
[11] F. Brezzi, L. P. Franca, T. J. R. Hughes, and A. Russo, \( b=∫ g \)., Comput. Methods Appl. Mech. Engrg., (1997), pp. 329–339.
[12] F. Brezzi, G. Manzini, D. Marini, P. Pietra, and A. Russo, Discontinuous finite elements for diffusion problems, in Atti Convegno in onore di F. Brioschi, Milan, Italy, 1997, pp. 197–217.
[13] F. Brezzi, G. Manzini, D. Marini, P. Pietra, and A. Russo, Discontinuous Galerkin approximations for elliptic problems, Numer. Methods Partial Differential Equations, 16 (2000), pp. 365–378. · Zbl 0957.65099
[14] A. N. Brooks and T. J. R. Hughes, Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg., 32 (1982), pp. 199–259. · Zbl 0497.76041
[15] A. Buffa, T. J. R. Hughes, and G. Sangalli, Analysis of a multiscale discontinuous Galerkin method for convection-diffusion problems, SIAM J. Numer. Anal., 44 (2006), pp. 1420–1440. · Zbl 1153.76038
[16] V. M. Calo, Residual–Based Multiscale Turbulence Modeling: Finite Volume Simulations of Bypass Transition, Ph.D. thesis, Stanford University, Stanford, CA, 2004.
[17] B. Cockburn, G. E. Karniadakis, and C.-W. Shu, eds., Discontinuous Galerkin Methods: Theory, Computation and Applications, Springer, Berlin, 11th ed., 2000.
[18] B. Cockburn, J. Gopalakrishnan, and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47 (2009), pp. 1319–1365. · Zbl 1205.65312
[19] B. Cockburn, G. Kanschat, and D. Schötzau, The local discontinuous Galerkin method for linearized incompressible fluid flow: A review, Comput. & Fluids, 34 (2005), pp. 491–506. · Zbl 1138.76382
[20] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), pp. 2440–2463. · Zbl 0927.65118
[21] C. Coley and J. A. Evans, Variational Multiscale Modeling with Discontinuous Subscales: Analysis and Application to Scalar Transport, arXiv preprint, arXiv:1705.00082 [math.NA], 2017.
[22] S. S. Collis, The DG/VMS method for unified turbulence simulation, in Proceedings of the 32nd AIAA Fluid Dynamics Conference and Exhibit, Reston, VA, 2002.
[23] S. S. Collis and S. Ramakrishnan, The local variational multiscale method, Third MIT Conference on Computational Fluid and Solid Dynamics, 2005.
[24] J. Donéa and A. Huerta, Finite Element Methods for Flow Problems, John Wiley & Sons, Hoboken, 2003.
[25] J. Douglas and T. Dupont, Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods, Springer, Berlin, 1976, pp. 207–216.
[26] R. Hartmann, Numerical Analysis of Higher Order Discontinuous Galerkin Finite Element Methods, in CFD - ADIGMA Course on Very High Order Discretization Methods, Von Karman Institute for Fluid Dynamics, Belgium, 2008, pp. 1–107.
[27] M.-C. Hsu and Y. Bazilevs, Fluid–structure interaction modeling of wind turbines: Simulating the full machine, Comput. Mech., 50 (2012), pp. 821–833. · Zbl 1311.74038
[28] M.-C. Hsu, D. Kamensky, Y. Bazilevs, M. S. Sacks, and T. J. R. Hughes, Fluid–structure interaction analysis of bioprosthetic heart valves: Significance of arterial wall deformation, Comput. Mech., 54 (2014), pp. 1055–1071. · Zbl 1311.74039
[29] A. Huerta, A. Angeloski, X. Roca, and J. Peraire, Efficiency of high-order elements for continuous and discontinuous Galerkin methods, Int. J. Numer. Methods Eng., 96 (2013), pp. 529–560. · Zbl 1352.65512
[30] T. J. Hughes and A. A. Oberai, Calculation of shear stresses in the Fourier-Galerkin formulation of turbulent channel flows: Projection, the Dirichlet filter and conservation, J. Comput. Phys., 188 (2003), pp. 281–295. · Zbl 1020.76025
[31] T. J. R. Hughes, Multiscale phenomena: Green’s functions, the Dirichlet–to–Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg., 127 (1995), pp. 387–401. · Zbl 0866.76044
[32] T. J. R. Hughes, G. R. Feijóo, L. Mazzei, and J.-B. Quincy, The variational multiscale method – a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg., 166 (1998), pp. 3–24. · Zbl 1017.65525
[33] T. J. R. Hughes, L. P. Franca, and G. M. Hulbert, A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least–squares method for advective–diffusive equations, Comput. Methods Appl. Mech. Engrg., 73 (1989), pp. 173–189. · Zbl 0697.76100
[34] T. J. R. Hughes, L. Mazzei, and K. E. Jansen, Large eddy simulation and the variational multiscale method, Comput. Vis. Sci., 3 (2000), pp. 47–59. · Zbl 0998.76040
[35] T. J. R. Hughes, L. Mazzei, A. A. Oberai, and A. A. Wray, The multiscale formulation of large eddy simulation: Decay of homogeneous isotropic turbulence, Phys. Fluids, 13 (2001), pp. 505–512. · Zbl 1184.76236
[36] T. J. R. Hughes, A. A. Oberai, and L. Mazzei, Large eddy simulation of turbulent channel flows by the variational multiscale method, Phys. Fluids, 13 (2001), pp. 1784–1799. · Zbl 1184.76237
[37] T. J. R. Hughes, G. Scovazzi, P. B. Bochev, and A. Buffa, A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 2761–2787. · Zbl 1124.76027
[38] T. J. R. Hughes, G. Scovazzi, and L. P. Franca, Multiscale and stabilized methods, in Encyclopedia of Computational Mechanics, E. Stein, R. De Borst, and T. J. R. Hughes, eds., John Wiley & Sons, Haboken, NJ, 2004.
[39] T. J. R. Hughes and J. R. Stewart, A space–time formulation for multiscale phenomena, J. Comput. Appl. Math., 74 (1996), pp. 217–229. · Zbl 0869.65061
[40] D. Kamensky, M.-C. Hsu, D. Schillinger, J. A. Evans, A. Aggarwal, Y. Bazilevs, M. S. Sacks, and T. J. R. Hughes, An immersogeometric variational framework for fluid–structure interaction: Application to bioprosthetic heart valves, Comput. Methods Appl. Mech. Engrg., 284 (2015), pp. 1005–1053.
[41] R. M. Kirby, S. J. Sherwin, and B. Cockburn, To CG or to HDG: A comparative study, J. Sci. Comput., 51 (2012), pp. 183–212. · Zbl 1244.65174
[42] C. Lehrenfeld and J. Schöberl, High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows, Comput. Methods Appl. Mech. Engrg., 307 (2016), pp. 339–361.
[43] D. Mavriplis, C. Nastase, K. Shahbazi, L. Wang, and N. Burgess, Progress in high-order discontinuous Galerkin methods for aerospace applications, Proceedings of the 47th AIAA Aerospace Sciences Meeting, Orlando, FL, 2009, 601.
[44] E. A. Munts, S. J. Hulshoff, and R. De Borst, A space-time variational multiscale discretization for LES, in Proceedings of the 34th AIAA Aerospace Sciences Meeting and Exhibit, Portland, OR, 2004. · Zbl 1120.76031
[45] J. Peraire and P.-O. Persson, The compact discontinuous Galerkin (CDG) method for elliptic problems, SIAM J. Sci. Comput., 30 (2008), pp. 1806–1824. · Zbl 1167.65436
[46] S. Ramakrishnan and S. S. Collis, Turbulence control simulation using the variational multiscale method, AIAA J., 42 (2004), pp. 745–753.
[47] W. H. Reed and T. R. Hill, Triangular mesh methods for the neutron transport equation, Proc. Amer. Nucl. Soc., 836 (1973), pp. 1–23.
[48] B. Rivière, M. F. Wheeler, and V. Girault, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I, Comput. Geosci., 3 (1999), pp. 337–360. · Zbl 0951.65108
[49] G. Sangalli, A discontinuous residual-free bubble method for advection-diffusion problems, J. Engrg. Math., 49 (2004), pp. 149–162. · Zbl 1041.76545
[50] D. Schillinger, I. Harari, M.-C. Hsu, D. Kamensky, S. K. F. Stoter, Y. Yu, and Y. Zhao, The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements, Comput. Methods Appl. Mech. Engrg., 309 (2016), pp. 625–652.
[51] S. Stoter, S. Turteltaub, S. Hulshoff, and D. Schillinger, A discontinuous Galerkin residual-based variational multiscale method for modeling subgrid-scale behavior of the viscous Burgers equation, Internat. J. Numer. Methods Fluids, to appear.
[52] T. E. Tezduyar, Stabilized finite element formulations for incompressible flow computations, Adv. Appl. Mech., 28 (1991), pp. 1–44. · Zbl 0747.76069
[53] Z. J. Wang, K. Fidkowski, R. Abgrall, F. Bassi, D. Caraeni, A. Cary, H. Deconinck, R. Hartmann, K. Hillewaert, H. T. Huynh, N. Kroll, G. May, P.-O. Persson, B. van Leer, and M. Visbal, High-order CFD methods: Current status and perspective, Internat. J. Numer. Methods Fluids, 72 (2013), pp. 811–845.
[54] F. Xu, D. Schillinger, D. Kamensky, V. Varduhn, C. Wang, and M.-C. Hsu, The tetrahedral finite cell method for fluids: Immersogeometric analysis of turbulent flow around complex geometries, Comput. & Fluids, 141 (2015), pp. 135–154. · Zbl 1390.76372
[55] Y. Xu and C. W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Commun. Comput. Phys., 7 (2010), pp. 1–46. · Zbl 1364.65205
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.