×

zbMATH — the first resource for mathematics

An artificial compression reduced order model. (English) Zbl 1434.76064

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
Software:
FEniCS; redbKIT; SyFi
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. Abdulle and O. Budáč, A Petrov-Galerkin reduced basis approximation of the Stokes equation in parameterized geometries, C. R. Math. Acad. Sci. Paris, 353 (2015), pp. 641-645. · Zbl 1320.76057
[2] I. Akhtar, A. H. Nayfeh, and C. J. Ribbens, On the stability and extension of reduced-order Galerkin models in incompressible flows, Theoret. Comput. Fluid Dyn., 23 (2009), pp. 213-237. · Zbl 1234.76040
[3] O. Axelsson and I. Gustafsson, Preconditioning and two-level multigrid methods of arbitrary degree of approximation, Math. Comp., 40 (1983), pp. 219-219, https://doi.org/10.1090/S0025-5718-1983-0679442-3. · Zbl 0511.65079
[4] M. Azaïez, T. Chacón Rebollo, and S. Rubino, Streamline Derivative Projection-Based POD-ROM for Convection-Dominated Flows. Part I: Numerical Analysis, arXiv e-prints, 2017, https://arxiv.org/abs/1711.09780.
[5] F. Ballarin, A. Manzoni, A. Quarteroni, and G. Rozza, Supremizer stabilization of POD-Galerkin approximation of parametrized steady incompressible Navier-Stokes equations, Internat. J. Numer. Methods Engng., 102 (2015), pp. 1136-1161. · Zbl 1352.76039
[6] M. Bergmann, C. H. Bruneau, and A. Iollo, Enablers for robust POD models, J. Comput. Phys., 228 (2009), pp. 516-538. · Zbl 1409.76099
[7] A. Caiazzo, T. Iliescu, V. John, and S. Schyschlowa, A numerical investigation of velocity-pressure reduced order models for incompressible flows, J. Comput. Phys., 259 (2014), pp. 598-616. · Zbl 1349.76050
[8] M. A. Case, V. J. Ervin, A. Linke, and L. G. Rebholz, A connection between Scott-Vogelius and grad-div stabilized Taylor-Hood FE approximations of the Navier-Stokes equations, SIAM J. Numer. Anal., 49 (2011), pp. 1461-1481, https://doi.org/10.1137/100794250. · Zbl 1244.76021
[9] J. de Frutos, B. García-Archilla, and J. Novo, Error analysis of projection methods for non inf-sup stable mixed finite elements: The Navier-Stokes equations, J. Sci. Comput., 74 (2018), pp. 426-455, https://doi.org/10.1007/s10915-017-0446-3. · Zbl 1404.65131
[10] V. DeCaria, W. Layton, and M. McLaughlin, A conservative, second order, unconditionally stable artificial compression method, Comput. Methods Appl. Mech. Engrg., 325 (2017), pp. 733-747.
[11] V. Eijkhout and P. Vassilevski, The role of the strengthened Cauchy-Buniakowskii-Schwarz inequality in multilevel methods, SIAM Rev., 33 (1991), pp. 405-419, https://doi.org/10.1137/1033098. · Zbl 0737.65026
[12] L. Fick, Y. Maday, A. T. Patera, and T. Taddei, A stabilized POD model for turbulent flows over a range of Reynolds numbers: Optimal parameter sampling and constrained projection, J. Comput. Phys., 371 (2018), pp. 214-243. · Zbl 1415.76387
[13] J. A. Fiordilino and M. McLaughlin, An Artificial Compressibility Ensemble Timestepping Algorithm for Flow Problems, preprint, http://arXiv:1712.06271, 2017.
[14] K. Friedrichs, On certain inequalities and characteristic value problems for analytic functions and for functions of two variables, Trans. Amer. Math. Soc., 41 (1937), pp. 321-364, https://doi.org/10.2307/1989786. · JFM 63.0364.01
[15] J. L. Guermond, Un résultat de convergence d’ordre deux en temps pour l’approximation des équations de Navier-Stokes par une technique de projection incrémentale, ESAIM Math. Model. Numer. Anal. Numérique, 33 (1999), pp. 169-189, http://www.numdam.org/item/M2AN_1999__33_1_169_0. · Zbl 0921.76123
[16] J. L. Guermond, P. Minev, and J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 6011-6045, https://doi.org/10.1016/j.cma.2005.10.010. · Zbl 1122.76072
[17] M. Gunzburger, N. Jiang, and M. Schneier, An ensemble-proper orthogonal decomposition method for the nonstationary Navier-Stokes equations, SIAM J. Numer. Anal., 55 (2017), pp. 286-304, https://doi.org/10.1137/16M1056444. · Zbl 1394.76067
[18] B. Haasdonk and M. Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations, ESAIM Math. Model. Numer. Anal., 42 (2008), pp. 277-302. · Zbl 1388.76177
[19] J. S. Hesthaven, G. Rozza, and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer, Cham, Switzerland, 2015. · Zbl 1329.65203
[20] J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19 (1982), pp. 275-311, https://doi.org/10.1137/0719018. · Zbl 0487.76035
[21] M. Hinze and S. Volkwein, Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: Error estimates and suboptimal control, in Dimension Reduction of Large-Scale Systems, P. Benner, D. C. Sorensen, and V. Mehrmann, eds., Springer, Cham, Switzerland, 2005, pp. 261-306. · Zbl 1079.65533
[22] P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, UK, 1998. · Zbl 0923.76002
[23] T. Iliescu and Z. Wang, Are the snapshot difference quotients needed in the proper orthogonal decomposition?, SIAM J. Sci. Comput., 36 (2014), pp. A1221-A1250. · Zbl 1297.65092
[24] T. Iliescu and Z. Wang, Variational multiscale proper orthogonal decomposition: Navier-Stokes equations, Numer. Methods Partial Differential Equations, 30 (2014), pp. 641-663. · Zbl 1452.76048
[25] N. Jiang and W. Layton, An algorithm for fast calculation of flow ensembles, Int. J. Uncertain. Quantif., 4 (2014), pp. 273-301. · Zbl 1301.65099
[26] V. John, Finite Element Methods for Incompressible Flow Problems, Springer Ser. Comput. Math., Springer International Publishing, New York, NY, 2016. · Zbl 1358.76003
[27] C. Jordan, Essai sur la geometrie a n dimensions, Amer. Math. Monthly, 3 (1875), pp. 103-174. · JFM 07.0457.01
[28] A. V. Knyazev and M. E. Argentati, Principal angles between subspaces in an A-based scalar product: Algorithms and perturbation estimates, SIAM J. Sci. Comput., 23 (2002), pp. 2008-2040, https://doi.org/10.1137/S1064827500377332. · Zbl 1018.65058
[29] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numer. Math., 90 (2001), pp. 117-148. · Zbl 1005.65112
[30] W. Layton, Introduction to the Numerical Analysis of Incompressible Viscous Flows, SIAM, Philadelphia, PA, 2008. · Zbl 1153.76002
[31] W. Layton, C. C. Manica, M. Neda, and L. G. Rebholz, Numerical analysis and computational testing of a high accuracy Leray-deconvolution model of turbulence, Numer. Methods Partial Differential Equations, 24 (2008), pp. 555-582. · Zbl 1191.76061
[32] A. Logg, K. Mardal, and G. Wells, Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, Springer Science & Business Media, New York, NY, 2012. · Zbl 1247.65105
[33] B. Mohammadi, Principal angles between subspaces and reduced order modelling accuracy in optimization, Struct. Multidiscip. Optim., 50 (2014), pp. 237-252, https://doi.org/10.1007/s00158-013-1043-1.
[34] M. Mohebujjaman, L. G. Rebholz, X. Xie, and T. Iliescu, Energy balance and mass conservation in reduced order models of fluid flows, J. Comput. Phys., 346 (2017), pp. 262-277. · Zbl 1378.76050
[35] B. R. Noack, M. Morzynski, and G. Tadmor, Reduced-Order Modelling for Flow Control, Springer-Verlag, Cham, Switzerland, 2011. · Zbl 1220.76008
[36] B. R. Noack, P. Papas, and P. A. Monkewitz, The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows, J. Fluid Mech., 523 (2005), pp. 339-365. · Zbl 1065.76102
[37] A. Prohl, Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations, Springer, New York, NY, 1997, https://doi.org/10.1007/978-3-663-11171-9. · Zbl 0874.76002
[38] A. Quarteroni, A. Manzoni, and F. Negri, Reduced Basis Methods for Partial Differential Equations: An Introduction, Springer, New York, NY, 2015. · Zbl 1337.65113
[39] T. C. Rebollo, E. D. Ávila, M. G. Marmol, F. Ballarin, and G. Rozza, On a certified Smagorinsky reduced basis turbulence model, SIAM J. Numer. Anal., 55 (2017), pp. 3047-3067, https://doi.org/10.1137/17M1118233. · Zbl 1380.65339
[40] S. Rubino, Numerical Analysis of a Projection-Based Stabilized POD-ROM for Incompressible Flows, preprint, http://arxiv.org/abs/1907.09213, 2019.
[41] J. R. Singler, New POD error expressions, error bounds, and asymptotic results for reduced order models of parabolic PDEs, SIAM J. Numer. Anal., 52 (2014), pp. 852-876. · Zbl 1298.65140
[42] L. Sirovich, Turbulence and the dynamics of coherent structures. Parts I-III, Quart. Appl. Math., 45 (1987), pp. 561-590. · Zbl 0676.76047
[43] G. Stabile and G. Rozza, Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier-Stokes equations, Comput. & Fluids, 173 (2018), pp. 273-284. · Zbl 1410.76264
[44] K. Veroy and A. T. Patera, Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds, Internat. J. Numer. Methods Fluids, 47 (2005), pp. 773-788. · Zbl 1134.76326
[45] S. Volkwein, Condition number of the stiffness matrix arising in POD Galerkin schemes for dynamical systems, PAMM. Proc. Appl. Math. Mech., 4 (2004), pp. 39-42, https://doi.org/10.1002/pamm.200410010. · Zbl 1354.65199
[46] J. Weller, E. Lombardi, M. Bergmann, and A. Iollo, Numerical methods for low-order modeling of fluid flows based on POD, Internat. J. Numer. Methods Fluids, 63 (2010), pp. 249-268. · Zbl 1423.76356
[47] D. Wells, Z. Wang, X. Xie, and T. Iliescu, An evolve-then-filter regularized reduced order model for convection-dominated flows, Internat. J. Numer. Methods Fluids, 84 (2017), pp. 598-–615.
[48] L. Wof and A. Shashua, Kernel principal angles for classification machines with applications to image sequence interpretation, in Proceedings of the 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, https://doi.org/10.1109/CVPR.2003.1211413.
[49] M. Yano, A space-time Petrov-Galerkin certified reduced basis method: Application to the Boussinesq equations, SIAM J. Sci. Comput., 36 (2014), pp. A232-A266. · Zbl 1288.35275
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.