zbMATH — the first resource for mathematics

A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem. (English) Zbl 1439.76057
Summary: The recently introduced divergence-conforming B-spline discretizations allow the construction of smooth discrete velocity-pressure pairs for viscous incompressible flows that are at the same time \(\inf - \sup\) stable and pointwise divergence-free. When applied to the discretized Stokes problem, these spaces generate a symmetric and indefinite saddle-point linear system. The iterative method of choice to solve such system is the Generalized Minimum Residual Method. This method lacks robustness, and one remedy is to use preconditioners. For linear systems of saddle-point type, a large family of preconditioners can be obtained by using a block factorization of the system. In this paper, we show how the nesting of “black-box” solvers and preconditioners can be put together in a block triangular strategy to build a scalable block preconditioner for the Stokes system discretized by divergence-conforming B-splines. Besides the known cavity flow problem, we used for benchmark flows defined on complex geometries: an eccentric annulus and hollow torus of an eccentric annular cross-section.
76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65D07 Numerical computation using splines
76D07 Stokes and related (Oseen, etc.) flows
Full Text: DOI
[1] Buffa, A.; Sangalli, G.; Vázquez, R., Isogeometric analysis in electromagnetics: B-splines approximation, Comput. Methods Appl. Mech. Eng., 199, 17-20, 1143-1152 (2010) · Zbl 1227.78026
[2] Buffa, A.; Rivas, J.; Sangalli, G.; Vázquez, R., Isogeometric discrete differential forms in three dimensions, SIAM J. Numer. Anal., 49, 2, 818-844 (2011) · Zbl 1225.65100
[3] Arnold, D. N.; Falk, R. S.; Winther, R., Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc., 47, 281-354 (2010) · Zbl 1207.65134
[4] Evans, J. A., Divergence-free B-spline discretizations for viscous flows (2011), ICES - UT Texas, (Ph.D. thesis)
[5] Evans, J. A.; Hughes, T. J.R, Isogeometric divergence-conforming B-splines for the Darcy-Stokes-Brinkman equations, Math. Models Methods Appl. Sci., 23, 04, 671-741 (2013) · Zbl 1355.76064
[6] Evans, J. A.; Hughes, T. J.R, Isogeometric divergence-conforming B-splines for the steady Navier-Stokes equations, Math. Models Methods Appl. Sci., 23, 08, 1421-1478 (2013) · Zbl 1383.76337
[7] Evans, J. A.; Hughes, T. J.R, Isogeometric divergence-conforming B-splines for the unsteady Navier-Stokes equations, J. Comput. Phys., 241, 0, 141-167 (2013) · Zbl 1349.76054
[8] Arrow, K. J.; Hurwicz, L.; Uzawa, H., Studies in linear and non-linear programming, Stanford Mathematical Studies in the Social Sciences (1964), Stanford University Press
[9] Elman, H. C.; Golub, G. H., Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 31, 6, 1645-1661 (1994) · Zbl 0815.65041
[10] Paige, C. C.; Saunders, M. A., Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12, 4, 617-629 (1975) · Zbl 0319.65025
[11] Saad, Y.; Schultz, M. H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 3, 856-869 (1986) · Zbl 0599.65018
[12] Axelsson, O., Iterative Solution Methods (1994), Cambridge University Press: Cambridge University Press New York, NY, USA · Zbl 0795.65014
[13] Elman, H. C., Multigrid and Krylov subspace methods for the discrete Stokes equations, Int. J. Numer. Methods Eng., 22, 8, 755-770 (1996), http://dx.doi.org/10.1002/(SICI)1097-0363(19960430)22:8<755::AID-FLD377>3.0.CO;2-1 · Zbl 0865.76078
[14] Elman, H. C.; Silvester, D. J.; Wathen, A. J., (Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, Numerical Mathematics and Scientific Computation (2005), Oxford University Press) · Zbl 1083.76001
[15] Benzi, M.; Golub, G. H.; Liesen, J., Numerical solution of saddle point problems, Acta Numer., 14, 1-137 (2005) · Zbl 1115.65034
[16] Rudi, J.; Malossi, A. C.I.; Isaac, T.; Stadler, G.; Gurnis, M.; Staar, P. W.J.; Ineichen, Y.; Bekas, C.; Curioni, A.; Ghattas, O., An extreme-scale implicit solver for complex PDEs: Highly heterogeneous flow in earth’s mantle, (Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC’15 (2015), ACM: ACM New York, NY, USA), 5:1-5:12, URL http://doi.acm.org/10.1145/2807591.2807675
[17] Cortes, A. M.A.; Coutinho, A. L.G. A.; Dalcin, L.; Calo, V. M., Performance evaluation of block-diagonal preconditioners for the divergence-conforming B-spline discretization of the Stokes system, J. Comput. Sci. (2015)
[18] Sarmiento, A.; Cortes, A. M.A.; Garcia, D.; Dalcin, L.; Collier, N.; Calo, V. M., PetIGA-MF: a multi-field high-performance toolbox of divergence-conforming B-splines, J. Comput. Sci. (2016)
[19] Dalcin, L.; Collier, N.; Vignal, P.; Cortes, A. M.A.; Calo, V. M., PetIGA: A framework for high-performance isogeometric analysis, Comput. Methods Appl. Mech. Eng., 308, 151-181 (2016)
[21] Vignal, P.; Sarmiento, A.; Côrtes, A. M.A.; Dalcin, L.; Calo, V. M., Coupling Navier-Stokes and Cahn-Hilliard equations in a two-dimensional annular flow configuration, Proced. Comput. Sci., 51, 934-943 (2015)
[22] Espath, L. F.R.; Sarmiento, A. F.; Vignal, P.; Varga, B. O.N.; Cortes, A. M.A.; Dalcin, L.; Calo, V. M., Energy exchange analysis in droplet dynamics via the Navier-Stokes-Cahn-Hilliard model, J. Fluid Mech., 797, 389-430 (2016) · Zbl 1422.76037
[23] Buffa, A.; de Falco, C.; Sangalli, G., Isogeometric analysis: stable elements for the 2D Stokes equation, Int. J. Numer. Methods Fluids, 65, 11-12, 1407-1422 (2011) · Zbl 1429.76044
[24] Piegl, Les A.; Tiller, W., The NURBS Book (1996), Springer · Zbl 0828.68118
[25] Cottrell, J. A.; Hughes, T. J.R; Bazilevs, Y., Isogeometric Analysis: Toward Integration of CAD and FEA (2009), Wiley · Zbl 1378.65009
[26] Hughes, T. J.R; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, cmame, 194, 4135-4195 (2005) · Zbl 1151.74419
[27] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, ESAIM: Math. Modelling Numer. Anal. - Modlisation Math. Anal. Numrique, 8, R2, 129-151 (1974) · Zbl 0338.90047
[28] Bazilevs, Y.; Michler, C.; Calo, V. M.; Hughes, T. J.R., Weak Dirichlet boundary conditions for wall-bounded turbulent flows, Comput. Methods Appl. Mech. Eng., 196, 49-52, 4853-4862 (2007) · Zbl 1173.76397
[29] Bazilevs, Y.; Hughes, T. J.R., Weak imposition of Dirichlet boundary conditions in fluid mechanics, Computers & Fluids, 36, 1, 12-26 (2007) · Zbl 1115.76040
[30] Bazilevs, Y.; Michler, C.; Calo, V. M.; Hughes, T. J.R, Isogeometric variational multiscale modeling of wall-bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes, Comput. Methods Appl. Mech. Eng., 199, 13-16, 780-790 (2010) · Zbl 1406.76023
[31] Embar, A.; Dolbow, J.; Harari, I., Imposing Dirichlet boundary conditions with Nitsche’s method and spline-based finite elements, Int. J. Numer. Methods Eng., 83, 7, 877-898 (2010) · Zbl 1197.74178
[32] Boffi, Daniele; Fortin, Michel; Brezzi, Franco, Mixed finite element methods and applications, Springer Series in Computational Mathematics (2013), Springer: Springer Berlin, Heidelberg · Zbl 1277.65092
[33] Glowinski, R.; Fortin, M., (Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems. Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, Studies in Mathematics and its Applications (2000), Elsevier Science)
[34] Wathen, A.; Silvester, D., Fast iterative solution of stabilised Stokes systems part I: using simple diagonal preconditioners, SIAM J. Numer. Anal., 30, 3, 630-649 (1993) · Zbl 0776.76024
[35] Simoncini, V., Block triangular preconditioners for symmetric saddle-point problems, Appl. Numer. Math., 49, 1, 63-80 (2004), Numerical Algorithms, Parallelism and Applications · Zbl 1053.65033
[36] Notay, Y., A new analysis of block preconditioners for saddle point problems, SIAM J. Matrix Anal. Appl., 35, 1, 143-173 (2014) · Zbl 1297.65034
[37] Amestoy, P. R.; Duff, I. S.; L’Excellent, Jean-Yves; Koster, J., A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl., 23, 1, 15-41 (2001) · Zbl 0992.65018
[38] Amestoy, P. R.; Guermouche, A.; L’Excellent, Jean-Yves; Pralet, S., Hybrid scheduling for the parallel solution of linear systems, Parallel Comput., 32, 2, 136-156 (2006)
[39] Collier, N.; Pardo, D.; Dalcin, L.; Paszynski, M.; Calo, V. M., The cost of continuity: A study of the performance of isogeometric finite elements using direct solvers, Comput. Methods Appl. Mech. Eng., 213-216, 0, 353-361 (2012) · Zbl 1243.65137
[40] Trottenberg, U.; Oosterlee, C. W.; Schuller, A., Multigrid (2001), Academic Press, Inc.: Academic Press, Inc. Orlando, FL, USA
[41] Adams, M., Evaluation of three unstructured multigrid methods on 3D finite element problems in solid mechanics, Int. J. Numer. Methods Eng., 55, 5, 519-534 (2002) · Zbl 1076.74547
[42] Saad, Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14, 2, 461-469 (1993) · Zbl 0780.65022
[43] Gee, M. W.; Siefert, C. M.; Hu, J. J.; Tuminaro, R. S.; Sala, M. G., ML 5.0 Smoothed Aggregation User’s Guide, Tech. Rep. SAND2006-2649 (2006), Sandia National Laboratories
[44] Henson, V. E.; Yang, U. M., BoomerAMG: A parallel algebraic multigrid solver and preconditioner, Appl. Numer. Math., 41, 1, 155-177 (2002), Developments and Trends in Iterative Methods for Large Systems of Equations - in memorium Rudiger Weiss · Zbl 0995.65128
[45] Collier, N.; Dalcin, L.; Pardo, D.; Calo, V., The cost of continuity: performance of iterative solvers on isogeometric finite elements, SIAM J. Sci. Comput., 35, 2, A767-A784 (2013) · Zbl 1266.65221
[47] Ballal, B. Y.; Rivlin, R. S., Flow of a Newtonian fluid between eccentric rotating cylinders: Inertial effects, Arch. Ration. Mech. Anal., 62, 3, 237-294 (1976) · Zbl 0354.76073
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.